Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/collectedmathema01smituoft I i ?^. COLLECTED MATHEMATICAL PAPERS OF HENRY J. S. SMITH i i £ondon HENRY FROWDE Oxford University Press Warehouse Amen Corner, E.C. Q\m Sort MACMILLAN & CO., 66 FIFTH AVENUE Ac^i^^r ^^a^^- MATIlEMArJCAI OF HENRY JOHN STE^^' . ^^... MA, ;te^ F.R.>. LATE SAVILIAN PROFESSOR OF RS i<D \A?ITM A VIATMPU. iTi-iCJ Qir\nUACMI -L'.' I T^p,^ /, t AT THE CLARKNDON PRKSS THE COLLECTED MATHEMATICAL PAPERS OF HENRY JOHN STEPHEN SMITH M.A., F.R.S. LATE SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD EDITED BY J. W. L. GLAISHER, Sc.D., F.R.S. FELLOW OF TRINITY COLLEGE, CAMBRIDGE WITH A MATHEMATICAL INTRODUCTION BY THE EDITOR, BIOGRAPHICAL SKETCHES AND A PORTRAIT IN TWO VOLUMES VOLUME I 3^^ lb M p' OXFORD AT THE CLARENDON PRESS 1894 O;cfot6 PRINTED AT THE CLARENDON PRESS BY HORACF HAUT, PBIMTER TO THE UNIVERSITY CONTENTS OF VOLUME I Biographical Sketch .... By De. Chaeles H. Peabson. Recollections of Henry J. S. Smith— By Pbofessor Jowett By Lord Bowen By Mr. J. L. Steachan-Davidson Note by Me. AirEED Robinson Introdl'ction to the Mathematical Papers By Db. J. W. L. Glaisheb. PAOE ix xxxvii xlvi li liv Ixi LIST OF PAPERS. I. On some of the Methods at present in use in Pure Geometry .... Transactions of the Ashmolean Society, Vol. II. No. xxv. Read Dec. 1, 1851. II. On some Geometrical Constructions ........ Cambridge and Dublin Mathematical Journal, Vol. VII. pp. 118-126. May, 1852. II I. De Compositione Numerorum Primorum formae 4A + 1 ex Duobus Quadratis . Crelle's Journal, Vol. L. pp. 91, 92. 1855. l\. On the History of the Researches of Mathematicians on the subject of the series of Prime Numbers .......•• Proceedings of the Ashmolean Society, Vol. III. No. xxxv. pp. 128-131. Read March 2, 1857. V. Report on the Theory of Numbers. Part I ...... • Report of the British Association for 1859, pp. 228-267. \'I. Report on the Theory of Numbers. Part II Report of the British Association for 1860, pp. 120-169. 1 25 33 35 88 93 vi ('ONTEXTS. PAOE 168 229 268 289 365 367 VTI. Rejjort on the Theory of Numbers. Part III Rt^port of the British Association for 1861, pp. 292-840. VIII. Report on the Theory of Numbers. Part IV Report of the Briti»h Association for 1S62, pp. 603-626. IX. Report on the Theory of Numbers. Part V Roport of the British Association for 1868, pp. 768-786. X. Report on the Theory of Numbers. Part VI Report of the British Association for 1865, pp. 822-376. XI. On Systems of Indeterminate Linear Equations Report of the British Association for 1860. Sectional Proceedings, p. 6. XII. On Systems of Linear Indeterminate Equations and Congruences PhiloMphical Transactions, Vol. CLI. pp. 298-826. Received Jan. 17 ; Read Jan. 31, 1861. XIII. On the Criterion of Resolubility in Integral Numbers of the Indeterminate Equation /= aa^ + <t'x"' + a"x"^ + 2 6i;'x" + 2 l>'xx"+ 2 6"x'a;=0 . . 410 Proceedings of the Royal Society, Vol. XIII. pp. 110, 111. Received Jan. 20 ; Read Jan. 28, 1864. XIV. On the Orders and Genera of Quadratic Forms containing more than Three Indeterminates ........... 412 Proceedings of tlie Royal Society, Vol. XIII. pp. 199-208. Received March 22 ; Read April 21, 1864. W. On Complex Binary Quadratic Forms 418 Proceedings of the Royal Society, Vol. XIII. pp. 278-298. Received May 18 ; Read June 16, 1864. XVI. On a Formula for the Multiplication of Four Theta Functions . . . 443 Proceedings of the London Mathematical Society, Vol. I. No. 8. pp. 8-14. Read May 21, 1866. XVII. On the Orders and Genera of Ternary Quadratic Forms .... 455 Philosophical Transactions, Vol. CLVII. pp. 255-298. Received Feb. 21 ; Read Feb. 27, 1867. XVni. On the Orders and Genera of Quadratic Forms containing more than Three Indeterminates 510 Proceedings of the Royal Society, Vol. XVI. pp. 197-208. Received Oct. 80 ; Read Dec. 6, 1867. XIX. On some Geometrical Constructions 524 Proceedings of the Loudon Mathematical Society, Vol. II. pp. 86-100. Read May 28, 1868. XX. Observatio Geometrica 541 Annali di Matcmatica, Ser. II. Vol. II. pp. 818-321. 1869. XXI. CJn the Focal Properties of nomographic Figures ..... 545 Proceedings of the London Mathematical Society, Vol. II. pp. 196-248. Read April 8, 1869. XXII. On the Focal Properties of Correlative Figures Proceedings of the London Mathematical Society, Vol. III. p. 12. Read Dec. 9, 1869. CONTENTS Vll VOLUME II PASS XXIII. Memoire sur Quelques Problemes Cubiques et Biquadi-atiques . . 1 Annali di Matematica, Ser. II. Vol. III. pp. 112-165, 218-242. Memoire Couronn6 par I'Academie Royaje des Sciences de Berlin, avec une moitie du prix Steiner en Juillet, 1868. XXIV. Arithmetical Notes 67 Proceedings of the London Mathematical Society, Vol. IV. pp. 236-258. The three papers which form these Notes were read on Jan. 9 and Feb. 13, 1873. XXV. On the Integration of Discontinuous Functions 86 Proceedings of the London Mathematical Society, Vol. VI. pp. 140-153. Read June 10, 1875. XXVI. On the Higher Singularities of Plane Curves 101 Proceedings of the London Mathematical Society, Vol. VI. pp. 153-182. Read June 10, 1875. XXVII. Mathematical Notes 132 Proceedings of the London Mathematical Society, Vol. VII. pp. 237, 238. Read Dec. 9, 1875. First printed in the Messenger of Mathematics, Vol. V. pp. 143, 144 (Jan. 1876). XXVIII. Note on Continued Fractions 135 Messenger of Mathematics, Ser. IL Vol. VI. pp. 1-14 (M.iy, 1876). XXIX. Note on the Theory of the Pellian Equation, and of Binary Quadratic Forms of a Positive Determinant . . . . . . . 148 Proceedings of the London Mathematical Society, Vol. VII. pp. 199-208. Read May 11, 1876. ^ XXX. On the Value of a Certain Arithmetical Determinant . . . . 161 Proceedings of the London Mathematical .Society, Vol. VII. pp. 208-212. Read May 11, 1876. XXXI. On the Present State and Prospects of Some Branches of Pure Mathe- matics ............ 166 Proceedings of the London Mathematical Society, Vol. VIII. pp. 6-29. Read Nov. 9, 1876. XXXII. On the Conditions of Perpendicularity in a Parallelepipedal System . 191 Proceedings of the London Mathematical Society, Vol. VIII. pp. 88-103. Read Dec. 14, 1876. XXXIII. On the Conditions of Perpendicularity in a Parallelepipedal System 213 Philosophical Magazine, Ser. V. Vol. IV. pp. 18-25. Read before the Crystallological Society, June 14, 1876. XXXIV. Sur les Integrales Elliptiques Completes 221 Atti della E. Accademia dei Lincei. Transunti, Ser. III. Vol. I. pp. 42-44. Read -Ian. 7, 1877. Vlll CONTENTS. PAttS 224 242 XXXV. M^moire sur les fiqiutions Modulaires Atti delU R. AocademiA doi Lincei. Memorie della cUsse di Scienze fisiche, mate- mstiche c naturalt Ser. lU. VoL I. pp. 186-149. Read Feb. 4, 1877. XXXVI. On the Singularities of the Modular Equations and Curves . Procvedings of the London Mathematical Society, Vol. IX. pp. 242-272. Read Feb. 14 and April 11, 1878. XXXVII. Note on a Modular Equation of the Transformation of the Third Order 274 Proceedings of the London Mathematical Society, Vol. X. pp. 87-91. Read Feb. 18, 1879. XXXVIII. Note on the Formula for the Multiplication of Four Theta Functions . 279 Proceedings of the London Mathematical Society, Vol. X. pp. 91-100. Read Feb. 18, 1879. XXXIX. De Fractionibus Quibusdam Continuis 287 Collectanea Mathematica (in memoriam Dominici Chelini), Milan, 1881, pp. 117-148. The paper is dated 1879. XL. On some Discontinuous Series considered by Riemann .... 312 Messenger of Mathematics, Ser. II. Vol. XI. pp. 1-11 (May, 1881). XLI. Notes on the Theory of Elliptic Transformation 821 Messenger of Mathematics, Ser. II. Vol. XII. pp. 49-99 (August-November, 1882). XLII. Notes on the Theory of Elliptic Transformation . . . . . 868 Messenger of Mathematics, Ser. II. Vol. XIII. pp. 1-54 (May-August, 1888). XLIII. Memoir on the Theta and Omega Functions 415 XLIV. M^moire sur la Repr6sentation des Nombres par des Sommes de Cinq Carres 623 M^moires pr^sent^s par divers sayants & I'Academie des Sciences de I'lnstitut National de France, ^ol. XXIX. APPENDIX. I. Address to the Mathematical and Physical Section of the Britisli Association at Bradford in 1873 681 II. Arithmetical Instruments 691 III. Geometrical Instruments and Models 698 IV. Introduction to the Mathematical Papers of William Kingdon Clifford . 71 1 BIOGEAPHICAL SKETCH I THE short record of Henry Smith's life, which I have compiled at the request of his sister, is chiefly based upon a Memoir by herself, which I was anxious to give in its entirety, but for which she has desired me to substitute my own words. I have to thank Professor Irving for a letter containing his recol- lections of Henry Smith at Balliol in his first years as Fellow and Lecturer, The Memoir and Letters from which I have worked are unfortunately most defective during the years of my own absence from England, 1871- 1883 ; and I must ask my readers to bear in mind under what disad- vantages I have attempted to perform a sacred duty. Happily, Henry Smith's character, about which there is really no difference of opinion, exhibits an unbroken continuity of growth. As a boy he seemed to have something of the mature wisdom of a man ; and to the day of his death he retained the simplicity and high spirits of a boy. My own estimate of him, based on the close intimacy of more than twenty years, represents, I hope and believe, what his friends thought and would wish said. To those who did not know him, it will perhaps appear that my judgment has been influenced by friendship. Those who knew him wUl notice points I have missed or excel- lences I have slurred, and will condemn my inadequacy. The original plan of this Memoir assumed that it would be supplemented by the publication of a large number of Henry Smith's letters. This was over-ruled in Oxford while I was in Australia, and cannot now be reverted to. Of these letters a few only have been published. Neither, unfortunately, X BIOGRAPHICAL SKETCH, have I leisure or strength to recast the Memoir altogether. I have done what I can in that direction, and can only hope that the difficulties under which I have worked may be borne in mind. CHARLES H. PEARSON*. Melbourne, 1888. London, 1893. HENRY JOHN STEPHEN SMITH was the second son of John Smith, a distinguished though short-lived Irish barrister, who, after graduating at Trinity College, Dublin, was law pupil, at the Temple, of Serjeant Henry John Stephen, the learned editor of Blackstone's Commentaries. Mr. John Smith went back to pi-actise in Dublin, and in 1818 married Mary Murphy. By this marriage he had four children — a daughter, who died of consumption in 1834 at the age of fifteen ; a son, Charles, who died also of consumption in 1843, being then a cadet at Addiscombe ; a daughter who still lives ; and the subject of this memoir, who was named after his father's old tutor, and who was born on the 2nd of November, 1826. Mr. Smith died in 1828, of abscess of the liver, and his widow was left for a time in very straitened circumstances. Fortunately, after delays which seemed interminable, the Courts affirmed the validity of a bequest of £10,000, which had been made to Mr. Smith by his cousin the Marchioness of Ormond, and which her husband disputed. With this money, and with that produced by the sale of a house which Mr. Smith had just built, the widow had wherewithal to provide adequately for her family ; and partly to escape from sorrowful asso- ciations, partly to secure for her children that good education which it had been their father's earnest wish they should receive, Mrs. Smith resolved within six months of her husband's death to pass over into England. The family wandered successively to the Isle of Man, 1829 ; to Harborne near Bu'mingham, 1829-30; to Leamington, 1830-31 ; and then to Ryde in the Isle of Wight, where nine or ten years were spent. Probably no widow left in charge of a young family could have been better fitted to train them for eminence in after life than was Mrs. Smith. A tall * Dr. Pearson's death has deprived this ' Sketch ' of the benefit of the author's final revision. BIOGRAPHICAL SKETCH. XI distinguished-looking lady, who retained the traces of great beauty to the day of her death, and who united a certain stateliness of manner and reserve of temperament to Irish ease and kindly charm of manner, she was also considerably more than an accomplished and clever woman, for she possessed powers and learning such as are rare even among able and learned men. Henry Smith inherited genius on both sides. He was a sickly child, and he was also short-sighted from a very early age, perhaps partly from being allowed to read too much when he was quite young. In 1831, when he was only four years old, he was able to follow English and French lessons. He also picked up an old Greek grammar of his mother's, rendered additionally formidable by contractions, and learned the alphabet, the nouns, the adjec- tives and the pronouns for his own pleasure. ' His practice,' says his sister, ' was to lay himself at full length on his stomach on the floor with the book he wished to study under his chin to suit his sight. When he was between seven and eight I remember Prideaux' Connection being for long an absorbing study.' As soon as his mother found out how marked his taste for language was, she took him into her own hands for the classics, and for the next six or seven years he owed all his chief training to her. In 1838 the pupil had got so far that his mother thought it desirable to call in other aid. She was fortunate enough to meet with a highly trained tutor in the person of Mr. R. Wheler Bush, who has put his recollections of Henry Smith on record in the following terms : * In the years 1838-39 Henry Smith, then a boy of eleven years of age, read with me for about nine months at Hyde, in the Isle of Wight. He had been previously taught by his widowed mother — a remarkably clever and highly educated woman. After reading with Henry Smith I had a large experience of boys during a head-mastership of more than thirty-three years, but I have often remarked that the brilliant talents of Henry Smith prevented me from ever being really astonished at the abilities of any subsequent pupil. His power of memory, quickness of perception, indefatigable diligence, and intuitive grasp of whatever he studied were very remarkable at that early age. What he got through during those few months, and the way in which he got through it, have never ceased to surprise me. From a record which I have before me I see that during that short time he read all Thucydides, Sophocles, and Sallust, twelve books of Tacitus, the greater part of Horace, Juvenal, Persius, and b2 jjjj BIOGRAPHICAL SKETCH. several plays of iEschylus and Euripides. I see also that he got up six books «)f Euclid, and algebra to simple equations ; that he read a considerable quantity of Hebrew ; and that, among other things, he learnt all the Odes of Horace by heart. I could scarcely understand at the time how he contrived at his early age to translate so well and so accurately the most difficult speeches of Thucydides, without note or comment to guide him. He was a deeply interesting boy, singularly modest, lovable, and affectionate.' {Times, Feb. 12, 1883.) Scarcely less valuable for the boy's development were the abundant leisure that he enjoyed, and the comparative isolation. His lessons never occupied more than five hours a day, and the obligatory ' constitutional ' was only of an hour. During the rest of the time the brothers and sisters were turned out to play by themselves. Their story books Avere limited to Robinson Crusoe, Evenings at Home, Sanclford and Merton, and Miss Edgeworth's Frank ; their toys consisted of hoops and tops, and one or two dissected games. They grew up like the young Brontes, in a world of their own, improvising plays from Robinson Crusoe or combats from Homer. In one of these fights Henry had his finger badly hurt by an arrow from the bow of Achilles, his elder brother, and the surgeon's aid had to be called in. These amusements could not occupy their whole leisure. In idle hours the children became diligent students of animal and insect life, leai'ning much about the habits of bees and ants and spiders and wood-lice and garden moths. They were directed in these pursuits by two books, Insect Architecture and Insect Transformxitions, fi^om the Library of Entertaining and Useful Knowledge, and assisted in them by two neighbours, a lady who was something of a botanist and a conchologist, and a Mr. Jacques, who had some knowledge of chemistry. That the interest they took in these matters was more than cursory seems proved by the fact that they supplied Dr. Blomfield, who was engaged on a Flora of the Isle of Wight, with several new homes of rare plants. In 1839 Mr. Bush was called away to a head-mastership. It proved difficult to supply his place, though an excellent mathematical master was found at Newport, who came over twice a week and carried his pupils through the advanced parts of Arithmetic, elementary Algebra, and Euclid. Henry continued to be a very docile pupil, sometimes asking, when he received an order which displeased him, whether he was ' forced' to obey it, but never demurring if he understood that obedience was required. In 1840, however, he lost his chief fellow-student. BIOGRAPHICAL SKETCH. XIU through his elder brother going to Addiscombe, and Mrs. Smith decided on moving to Oxford, where it was certain that better teaching could be found than in the Isle of Wight. The Oxford of those days was comparatively a small place. Resident professors, married tutors, and married fellows were almost or quite unknown, while the Heads of Houses, then the governing body of the Univer- sity, formed a little society of their own. Consequently, the widow lived in comparative solitude, though even so she could not avoid hearing something of the war of opinion that was beginning : of the angry opposition provoked by Tract 90, of Newman's sermons, of the coalition of Evangelicals and Liberals against Puseyites, and now and again of the few Liberals who stood outside the strife of the Churches. Meanwhile she had been exceptionally fortunate in the tutor she secured for her son. The Rev. Henry Highton, Fellow of Queen's and then Curate of St. Ebbe's, was a sound though not a brilliant scholar, and a really good mathematician, far above the average of Oxford in those days. No one could be better fitted to develope Henry Smith's varied capacities, and in Mr. Highton's class-room Henry, for the first time, was able to measure himself with boys of his own age. In the summer of 1 841 Mr. Highton received the offer of a Mastership at Rugby, which at that time was chiefly valuable when a boarding- house was attached to it. Mr. Highton accepted the offer, which allowed of his maiTying, and proposed that he should take Henry Smith with him as his first boarder. Mrs. Smith agreed, and Henry was thus launched into school life under the most famous teacher of the day. Dr. Arnold. I have always regarded it as singularly fortunate for Henry Smith that he was at Rugby in its best days, and that he was not there long enough to acquire that part of its tone which was not generally popular. Whether that sweet buoyant nature, with its supreme sense of proportion, and lively humour, could ever have been really spoOed, made pedantic or harsh, is perhaps more than doubtful ; but I cannot doubt that the years of travel on the Continent, which two chances, that seemed unkindly, substituted for school and Oxford life, were really of the greatest use to the sufferer. He carried on his studies abroad less methodically, but quite as profitably, as he could have done at home ; he learned French, German, and Italian, and he gained some acquaintance with foreign ideas and methods. Meanwhile his first years at Rugby were certainly profitable to him. It was a rule of the school that no one should be in the Sixth Form until he was sixteen, and in deference to this rule xiv BIOGRAPHICAL SKETCH. Henry Smith was kept for a year doing work below his capacity in the Upper Fifth and the Twenty Form, though by a curious anomaly he was allowed to act as pnepostor in Mr. Highton's house, where he was the senior boy. His Report for the first half-year, which was spent in the Fifth Form, has been preserved ; and might have been written of him at almost any time : — ' Classics : In extent and variety of knowledge he is certainly the best in the Form, and he is particularly fortunate in combining accurate and literal construing with an excellent choice of English words. His composition also (though it has sometimes been careless) is spirited and clever. Mathematics : Very good. Modern Languages : He has made great progress in German, and is getting on very well. He has been late for morning prayers oftener than I like ; and I should wish him to get rid of a few trifling irregularities, such as occasional inattention at lesson and inaccuracy in saying his lines. G. E. L. Cotton, Master of the Fifth Form,' When in the Twenty he came under Mr. (afterwards Professor) Bonamy Price, con- fessedly the ablest teacher on the very able staff of which Rugby then boasted, and probably never surpassed as a teacher of classics. In the Midsummer exam- ination of that year, Henry Smith passed into the Sixth, and was accordingly entitled to bid the Doctor good-bye, A few days later he received a letter from Mr. Highton (June 12) : — ' You hardly supposed that when you bid Dr. Arnold " good-bye " on Friday it was for the last time. He was taken to his rest at six this morning. . . . You may imagine how the loss is felt here. It is almost as if a common parent were taken away. I felt it so quite myself.' The true education of a boy at a public school is even more in the play- ground than in the class-room. What Henry Smith was in this regard has not come down to me. Going to Rugby just before he left it, I remember to have heard ' Highton Smith,' as he was popularly called, spoken of with vague reverence for his great ability, but in no other way. Nevertheless there are in- dications from reports and letters that he was abundantly capable of healthy enjoyment, and not merely what the Rugbeans used to call a ' swat ' or book- worm. Mr. Highton twice reports of him in his first half-year that he was not working as hard as he ought ; and his sister says that ' he came home for his first holidays " astonishing " iis by the buoyancy of his spirits and even more by a propensity for " grub," unknown to the ascetic days of his childhood. By one who learned so easily as he did, a little idleness was easily made up for.' In the examination of June 1843 he obtained a Junior Scholai-ship, being ineligible for BIOGRAPHICAL SKETCH. XV the University Scholarship because he had not been three years in the school. In July 1843 the new head master, Dr. Tait, wrote to Mrs. Smith to say : — ' There is no young man in the Sixth Form from whose abilities I am led to expect more than from him ; and I have formed a very high opinion of his character and conduct generally.' Rugby however was not to keep him. In September 1843 his elder brother Charles died of rapid consumption, and the uncle, who was also guardian and adviser, declined under these circumstances to consent to Henry's remaining any longer at a school in a bleak part of England. The boy bore the blow to his ambition with his unvarying sweetness, and wintered with his family at Nice, while he spent the following summer by the Lake of Lucerne. These were months of steady reading, though his books were few, and he was even unpro- vided with a Greek lexicon. In the autumn of 1844 he went back to his old friend, Mr. Highton, for a month, that he might be ' coached ' for the Balliol scholarship. He won it easily, and, as he was not to go into residence till Easter, went back to join his family at Rome. His journey was a series of disasters. He missed the mail-boat at Dover, had his pocket picked at Paris, and, even after pledging his books, could only muster funds enough to carry him in the roughest way to Rome. This misadventure involved a journey of seventy hours, on the outside of a diligence during a severe frost, to Marseilles ; and, after a third- class passage from that port to Civita Vecchia, he arrived in Rome with both his feet frost-bitten, and was laid up for a long time. Presently came an attack of small-pox. ' All the same,' writes his sister, ' the winter was a time of intense enjoyment, and a gathering and growing time.' By Easter he was well enough to go to Oxford, and spend his first term there. When the Long Vacation came Henry Smith rejoined his mother and sister in Italy. Unfortunately they arranged to spend the summer at Frascati in the Alban Hills, and Henry soon became languid and ailing, and at last ill enough to need a doctor. The doctor who came, an Italian physician of eminence, declared after three weeks that his patient was undoubtedly consumptive, and ordered him to the sea at Naples by way of the Pontine Marshes. Even in their alarm the family were discreet enough to substitute the hill route for the deadly road along the plains ; but this involved a four days' journey, during which the sufferer became delirious, and, when Naples was at length reached, the English doctors had aU left the city. One however was to be found at Castellamare, and he, Xvi BIOGRAPHICAL SKETCH. when he was called in, declared that the disease was nothing but long-neglected malaria, which an ordinary Italian doctor should have recognised. It was now- thought right to revert to the use of strong tonics. Severe inflammatory attacks subsequently came on. These have been, since then, attributed to the presence of gall-stones, which may have possibly laid the foundation of his latest illness. Moreover, with spring (1845), the malaria itself returned, and it became necessary to leave the South. He himself at a later time described his illness to me as a sort of euthanasia, in which he seemed to be gliding painlessly out of life. The sister who helj>ed to nurse him remembers that he was too weak even to put up his glass that he might look at an eruption of Mount Vesuvius. Still he was able to enjoy being read aloud to, and his mother used to read to him incessantly from English newspapers and standard authors, but especially from the Latin and Greek classics, while on Sundays he would honour the day by forbearing to cor- rect or even to shudder at a false quantity. The move from Naples was to Wiesbaden, and there the waters restored him to comparative health. It was thought better however that he should not return to England, and accordingly the next winter (1845-6) was spent at Paris, where Henry Smith attended several of the courses at the College of France or the Sorbonne, and derived especial advantages from the lectures of Arago and of Milne Edwards. By this time his strength was thoroughly re-established, and, though he went a second time to Wiesbaden in the summer of 1847, he had already resumed work at Oxford (Easter, 1847), and never afterwards needed to suspend it. His health, as I remember it for more than twenty years of unbroken intimacy, during which I was constantly seeing him, was always good, though never what could be called robust. He suffered especially as a young man from weak eyes ; and he had to 1)6 a little careful of himself in diet and exercise ; but he was rarely depressed, and he habitually worked beyond what most men could have endured without breaking down. There was one attack of low fever in 1856, the result of course of the earlier Roman fever, in consequence of which he was ordered to ride, and the obligation to take horse exercise was undoubtedly very good for him, and contributed a great deal to his enjoyment of life. The Oxford into which Henry Smith was now thrown had almost recovered from the strong ferment which ended in Newman's going over to the Church of Rome. The leaders of the High Church party had either followed their captain, like Christie and Bowles, or had satisfied themselves, like Mark Pattison, that BIOGRAPHICAL SKETCH. XVU Protestantism does not admit of a divided allegiance. There were still High Church cliques among the undergraduates, such as Newman has sketched with caustic subtlety in Loss and Gain, which discussed Church matters from the Anglo-Catholic point of view, but they rarely got beyond a mild dilettanteism. Even this was not always treated with a proper tolerance. I remember a de- bating society of young Churchmen, which so irritated the Protestantism or the common sense of a rather sporting College by carrying a resolution that ' St. Augustine's interference with the British Church was uncatholic and uncalled for,' that at its next meeting the orators were dispersed by the agency of hot pepper thrown into the room, and saluted with a baptism little short of total im- mersion as they left the quad. Of the fast men of that day, it need only be said that they have been inimitably limned for good and bad in Tom Brown at Oxford. Outside these two sets, which have perhaps attracted more attention than they deserve, and also outside the common and obscure men, were the abler young men of the University, Conservative or Liberal in their politics, as tempera- ment or training determined, but mostly with a wholesome share of English secularism, and neither High Church, except in rare instances, nor aggressively Protestant, nor to any appreciable extent Freethinkers. Lord Salisbury, Lord Kimberley, Lord Brabourne, Sir M. E. Grant Duff among politicians ; Goldwin Smith, Sellar, Grant, Sandars, Poste, and Conington among men of letters or scholars ; Spottiswoode and RoUeston among men of science ; Chitty among judges ; Sandford and Ducane among officials, were some of the Oxford men of Henry Smith's day, and with most of these he was more or less intimate at some time, while Grant Duff and Conington were among his dearest friends. Whatever time or thought men of this type could spare from work for the schools, was divided between politics and literature ; and Henry Smith's Univer- sity letters are a singularly faithful reflex of the spirit of the period. They are more mature and temperate than perhaps any one but himself could have written, but they show the enthusiasm for intellectual eminence which is the salt of Oxford life ; and the admiration evinced for Mill, and the praise, however qualified, of Robert Chambers, are evidence that the writer was already to be numbered among the few on whom Carlyle had no hold. Of Henry Smith's Oxford career it may briefly be noted that in 1848 he won the Ireland University Scholarship, the blue ribbon of classical scholars ; was a double first-class in the Lent Term of 1849 ; was elected Fellow of Balliol in Xviii BIOGRAPHICAL SKETCH. November 1849 ; and gained the Senior Mathematical Scholarship in 1851. He was unable through absence to stand for the Hertford Scholarship, which falls to the best Latin scholar of the year ; or for the Junior Mathematical ; and he was beaten for the Senior Mathematical Scholarship in 1850, the first time that he stood for it, by Mr. Ashpitel of Brasenose ; the single defeat of the kind, I believe, which Henry Smith sustained. Balliol College, to which Henry Smith belonged, was far away the best in the University during the time of his residence, and for some years afterwards. A variety of circumstances had contributed to build up its pre-eminence. The first cause was the far-sighted integrity of the old Master, Dr. Jenkyns, who was almost singular among the Heads of his day in regarding it as the first duty of a College to promote intellectual distinction, and who waged an incessant war with privilege, abolishing gentlemen-commoners and throwing open close endowments as far as he legally could. Dr. Jenkyns could not have done much single-handed, but he gradually found or created men, often no doubt abler than himself, who were glad to carry on his work in the same spirit ; and the late Master of Balliol, Mr. Jowett, then one of the tutors, was undoubtedly the soul of the College during the whole time of Henry Smith's connection with it. At the time of Henry Smith's election, the College wanted a mathematical lecturer. There is no doubt, I think, that he was chosen in the well-warranted expectation that he would consent to reside and lecture. In this way began his own lifelong union with Oxford, for untU then he had been a mere bird of passage. Having once decided to accept the office thrust upon him, he gave himself up heart and soul to doing his work well. It was a common story in Oxford at that time that Henry Smith, being uncertain after he had taken his degree whether he should devote himself to classics or mathematics, had solved the doubt by tossing up a halfpenny. His sister remembers how he actually expressed a wish that some one would do this for him. He was, in fact, the last man on earth to have committed any im- portant decisions to chance ; and he has himself told me that his choice was partly determined by the fact that having at that time weak sight he found he could do more work in thinking out problems than in any other way without usmg his eyes. The decisive reason was of course a pre-eminent genius for mathematics — the born aptitude that is itself fate — and the cause why the determination was made at that particular time may have been this offer of BIOGRAPHICAL SKETCH. XIX a lectureship. Nevertheless the Oxford tradition is so far valuable as it testifies to the general belief that Henry Smith could have made his mark in any study he embraced. ' I do not know,' Professor Conington once said to me, ' what Henry Smith may be at the subjects of which he professes to know something ; but I never go to him about a matter of scholarship, in a line where he professes to know nothing, without learning more from him than I can get from any one else.' Once it seemed as if he would be attracted into chemistry. The College demanded of him that he should give chemical lectures (1853), and Henry Smith accordingly became a pupil under Professor Story-Maskelyne, who then occupied a laboratory under the Ashmolean Museum and gave instruction in chemical analysis. Here H. Smith showed that in delicate manipulation and in accuracy of work he possessed a sort of instinctive faculty. A lifelong friendship grew out of the hoiu"S spent in this way ; although the demands of a new and engrossing science on his time were too great to permit his sacrificing to chemistry the many other important subjects and duties that filled up his life. Even then, however, I remember, his idea was to seek numerical relations connecting the atomic weights of the elements and some mathematical basis for their various properties*, so that we might anticipate experiments by the operations of the mind — an ambition which was very interesting to Alexander von Humboldt, when Sir M. E. (then Mr.) Grant Duff told him of it. Ultimately Henry Smith of course found that science is too jealous a mistress to admit of a divided allegiance ; and, though his reading was always wide and various and singularly weU digested, he practically devoted himself to mathematics, and as I understand to two or three great subjects with which his name will always be associated, the Theory of Numbers, the Theory of Elliptic Functions, and certain new processes of Geometry. One point for which the generations younger than Henry Smith and John Conington will always remember them gratefully was the way in which they mingled in undergraduate society. The distinctions of academical rank were at that time rather jealously marked in Oxford. If the tutors and fellows were * His conviction that such a numerical and mathematical basis underlay the phenomena of chemistry was even stronger in the case of crystals. At my suggestion he undertook the discussion of the principles involved in the parallelism of zone-axes and face-normals in a crystal system with rational indices ; the results of which were given by him to the London Mathematical Society in vol. viii of its Proceedings. — N. S. M. C 2 XX BIOGRAPHICAL SKETCH. not, like the Heads of Colleges, in inaccessible isolation, they were scarcely better known to us through the formal breakfast parties which were submitted to on both sides as a very irksome duty. Here and there a man, like the admirable Charles Marriott of Oriel, made it a duty to invite young men, in order that they might feel at liberty to go to him if they needed religious advice, which was never obtruded ; but men of this stamp were never, as far as I remember, more than genial hosts. Henry Smith and Conington, men of the most opposite temperament though devoted to one another, threw themselves with such unaffected simplicity into our interests and occupations, that we all came to regard them as personal friends, and to talk as freely before them as before one another. Looking back I can see that their position was that of very helpful and sympathetic seniors at a children's party, and I can conceive how Smith's playful sense of fun and Conington's grave humour must often have been tried by the obligation to treat our criticisms of men and things or our forecasts of the future seriously. That the intercourse was begun and carried on on their part from a conscientious desire — due partly to Ai-nold's teaching — to convey a serious interest into everyday life, I at least cannot doubt. As one who profited by the association let me record that meetings of this kind were not only the most pleasurable part of a chequered Oxford life to many of us, but unquestionably did more to stimulate thought and form character than the more formal influences of the chapel and the lecture-room. A letter from one of Henry Smith's old pupils, Professor Irving, of Melbourne, will complete the description of this part of his career, and speaks with authority on some matters which I only knew from report. Melbourne, let September, 1888. Mt dear Pearsox, It is by no means an easy task to carry back the memory nearly forty years, and recall scattered reminiscencea of one from whom you have been altogether parted, with whom you have not even kept up communication by letter for more than thirty years. Yet such in the old Oxford days was my affection for, and so highly have I ever honoured him whom I was then proud to call my friend, that I must accede to your request and do what little I can towards your presentment to the world of Henry Smith, Scholar and Mathematician. My first introduction to him was, I think, in my Freshman's Term at Oxford, Michael- mas 1849, the term in which he gained his Fellowship at Balliol ; but our mtimacy really BIOGRAPHICAL SKETCH. XXI dated from the beginning of 1850, when he became our Mathematical Tutor. From the first there had been this link between us, that I had inherited the little third-floor rooms in the inner quad of Balliol, which had belonged to Scholar Smith, as he was called in those days to distinguish him from sundry others of the same patronymic. I remained his pupil in Mathematics throughout my undergraduate time, and was able to do him the credit of winning the Junior Mathematical University Scholarship in 1850. Of his power as a Class Teacher I cannot speak, as I was almost alone with him : but my individual experience was that as a Teacher he was all a learner could desire, most patient with all one's difficulties, most clear and full in his explanations. But he was too kind to me. He sympathized strongly with my disappointment when the authorities refused me permission to compete for the Ireland in the spring of 1850, and knowing how keen my desire was to win that distinction in the following years, he did not force me to work at Mathematics as I ought to have done, and so I failed to do justice to his teaching, and attain his own class in the Final Honour Schools. Through Smith I made another valued friendship, that of John Conington, Professor of Latin, and might have gained access to the intellectual circle in which they moved, and of which they were such brilliant ornaments. But I must honestly confess that my own work for the Schools was sufiicient mental exercise for me, and that I sought my relaxation, not in other spheres of thought, but on the river. Yet in our frequent intercourse there was quite opportunity enough for me to learn and to appreciate the manysidedness of Henry Smith's mind. All of my generation were prepared to look up to and to admire one who had done so brilliantly as he had in Oxford : and when you came to know him personally you could not look upon that splendid fore- head of his without assurance of the powerful intellect it betokened : you could not con- verse with him without realising that he was one of whom it might be said that ' omne scibile novit.' And what he knew, he knew, not as so much stored up learning to be brought forth as required, but he had made it all his own, he had thought as well as read. Still with all his vast erudition, and his great intellectual power, he was the humblest, the gentlest of men. Ignorance, even if pretentious, was not to him something to be crushed with sledgehammer Johnsonian blows, but a thing to be pitied and kindly enlightened. In fact were I asked to select his peculiar moral characteristic, I should say he was the most gracious man I ever knew. Heading over these lines, I recognize with regret how very imperfectly are therein ex- pressed the love and admiration I felt for him, how feebly they serve to set him forth. Yet inadequate though they are, they may help somewhat. If I cannot do better — ' His saltem accumulem donis, et fungar inani Munere.' I am, my dear Pearson, Yours ever sincerely, M. H. IRVING. Xxii BIOGRAPHICAL SKETCH. In 1855 some of the abler Oxford and Cambridge men determined to publish a yearly volmne of Oxford and Cambridge Essays, and Oxford led the way, T. C. Sandars, I believe, being the guiding spirit and editor. Henry Smith contributed an essay on the Plurality of Worlds to this publication. He took for his theme Whewell's then unacknowledged book on that subject, and Sir David Brewster's fiery answer, ' More worlds than one, the Creed of the Philosopher and the Hope of the Christian.' The subject was a fascinating one, and Heniy Smith was in many respects admirably calculated to do it justice. He wrote as simply and clearly as Herschell or Tyndall, and he was skilful in dissecting arguments of every kind with faultless impartiality, so as to reduce them to their absolute value or no-value. A single passage will serve as an illustration of his method. Whewell had argued in the spirit of Lucretius that there was, so to speak, a law of waste traceable in the Divine economy, and supported this position by facts from the origin of species. ' If we found,' Henry Smith remarks, ' that Jupiter's four seasons differed so slightly from one another that they hardly deserved the name, and that they could not be conceived to be of any use to his hypothetical inhabitants, we should be reminded by those "rudimentary seasons" of the osteo- logical facts on which the essayist dwells so much, of the rudimentary fingers in the hoof of a horse, or the rudimentary paws with which a snake is said to be en- dowed. But the one thing we should not be prepared to find would be a wasted, imperfect, uninhabitable planet. We should know of no facts in Zoology with which to compare such an occurrence. The crust of our earth is filled with the remains of departed life, but we find not a vestige of imperfect attempts, of forms moulded after the vertebrate type, and yet incapable of animation.' It wlU be remembered that both essayist and critic wrote in the days before Darwin, and that Henry Smith had never made any special study of comparative anatomy. With this allowance it must, I think, be recognised that Whewell's conception of a general law producing a single successful result and failing in every other case was substantially and hopelessly wrong, and that his critic's conception of ' a law uniformly asserted in a multitude of individual cases, and uniformly pro- ductive of variously perfect results,' was a singularly correct expression of all that science was then able to teach. There are passages of characteristic irony scattered through the essay. We are told that in one of Plutarch's Dialogues, ' the lunar world is connected with the future destiny of the human soul, after a manner which, we conceive, BIOGRAPHICAL SKETCH. XX 111 Sir David Brewster would allow to be highly creditable for a heathen, and on the whole corroborative of his own opinion.' Of the literary form of Whewell's essay it is said : ' In the dialogue at the beginning of the essay the earlier letters of the alphabet, who appear as the objectors, conduct themselves so much like simpletons that one wonders at their being thought worthy of so long an inter- view with the enlightened Z.' Of theological arguments intruded into the domain of science it is observed : ' We cannot imagine a more painful spectacle of human presumption than that which would be afibrded by a man who should sit down to arrange " in a satisfactory way " a scheme for the extension of Divine mercy to some distant planet, and who, when he found " great diffi- culty in conceiving" such an extension of the Divine attribute, instead of desisting from his vain attempt, should go a step further still, and infer that no such scheme can exist because he fails to discover a modus operandi for it.' StiU, though the article on the ' Plurality of Worlds ' was read with pleasure and spoken of with esteem, its success was not so unquestionable as to tempt its author into larger literary work ; and his only other contributions of any length to English prose are, I believe, a review of Mr. Freeman's Federal Government, which he wrote for the National Review in 1864, and the Memoir of Professor Conington, which was prefixed to the volumes of his works in 1871, and which is a very perfect example of skill in recording a quiet life so as to invest it with interest. It would have been a great misfortune for science if a man capable of enlarging its boundaries had wasted his powers upon mere criticism or exposition ; yet, considering Henry Smith's unambitious tempera- ment, which made him careless of personal fame, and his invariable readiness to oblige friends, I cannot doubt that he might have been seduced into Quarterly Reviewing or some other form of ephemeral literature if he had possessed some of the minor qualifications of a journalist. The fault of his argumentative writing is a disposition to hold the balance and to avoid summing up ; and it is in keeping with this quality that his style, though it has the subflavour of irony and the point inseparable from lucid concentration, is not epigrammatic or what would be called strong. The writer's tenderness of disposition had something to do with this characteristic. He who as a boy of fifteen had stopped himself in a caustic criticism in a private letter because the subject of it was ' somebody's bairn ' carried the same thought for others into his words and dealings through Xxiv BIOGRAPHICAL SKETCH. life. Keserve on matters that lay near the heart was another modifying influence. To many men, the scholar who identified himself with every movement for religious or intellectual or political freedom in the University was still more or less a sphinx because he never propounded his convictions as a topic for conversation. The world that reads is rather like the world that listens to a platform orator. It likes its instructors to be positive, even where they cannot be certain, and to have its conclusions presented to it in the form of short and simple aphorisms, which it may swallow and retain without trouble. Henry Smith could not have attained to this ideal, and it is matter of some satisfaction that he did not aspire to it. In 1857 Mrs. Smith died. The attachment of mother and son to one another had been deeper than is common, and the course of their lives had drawn them nearer together than can often be the case. It was now arranged that Miss Smith should keep house with her brother in Oxford, the two spending the term together, and each being allowed complete liberty of move- ment during the vacations. I cannot doubt that this arrangement contributed very much to Henry Smith's happiness. He was eminently domestic and hospitable, and having the cares of household life taken off his hands, and being supplemented by one who was almost another self, was able to fill his house with friends, who were certain of an Irish welcome, however unseasonably they might arrive to ask for a dinner or a bed. He was also able under his own roof to gratify his passion for pets — Persian cats of distinction and two aristo- cratic dogs — to which there are frequent allusions in his letters. During the vacations he often visited the Continent, going once to Sweden and Norway ; more than once to North Italy; to Spain with Grant Duff in 1864 ; and to Greece in 1872 with Mr. and Mrs. Grant Duff. From time to time he paid visits to an old friend of the family. Miss Theodora Price, who had lived with his mother during the whole time of her widowhood, and who, on 'Mrs. Smith's death, established herself at Tunbridge Wells. For some years, too, Henry Smith was a prominent figure at the various meetings of the British Association in England, Scotland and Ireland. It will be seen that his life was in no sense that of a recluse ; and it may be added that he entered with zest into every form of social enjoyment in Oxford, from croquet parties and picnics to dinners. That the irregular, desultory life, with its frequent breaks, suited his health Is probable ; and, as he possessed a rare power of utilising stray hours so as never ( BIOGRAPHICAL SKETCH. XXV to intermit work altogether, even when his distractions were most numerous, it seemed possible that he might be among the singular few who have combined residence in an English University with unswerving devotion to the claims of abstract research. Fortune appeared to favour this anticipation. In 1860 Mr. Baden Powell, the Savilian Professor of Geometry, died, and Henry Smith became a candidate for the vacant post. Some years before, he had told a friend that to occupy this chair was the great ambition of his life. He said that the two Savilian Professorships were the most honourable offices in the University : they were open to the whole world of Mathematicians, and had usually been held by distinguished men. His wish was now to be gratified in the pleasantest way. Those who would naturally have been his rivals, the other Oxford mathema- ticians, were the first to draw back in his favour and sign a common testimonial to his pre-eminent claims*. I well remember the generous warmth with which two of the senior professors, Mr. Walker and Mr. Bartholomew Price, but especially the latter, expressed themselves to me at the time about Smith's undoubted genius and the chances that he would one day leave a name to be remembered beside those of Newton and Laplace. The electors for the chair chose him, as I have understood, without hesitation, taking the view that as no other Oxford man was a candidate, and as Henry Smith was pre-eminently qualified, it was needless to scrutinise the testi- monials of outsiders. He himself was a little troubled by a doubt whether the claims of an older man. Dr. Boole, of Queen's College, Cork, ought not to have received further consideration. That Henry Smith justified his electors by the magnificent work he did later on, is beyond question. He was also a very successful teacher, having what must be considered large classes in a University where mathematics have, at least in recent times, attracted comparatively few students. Passages in his letters prove how keenly he * I can express no opinion worth having on this subject ; but I see from a notice in the Academy (Feb. 17, 1883), written evidently by a personal friend, that much of the work given to the world in later years had been produced before he was thirty-five. ' He (Professor Smith) communicated at different times a good many notes and papers to the Mathematical Society, especially during his Presidency in 1874-76 ; and we believe that all the results he gave he had had in his possession for fifteen years,' His work on the Theory of Elliptic Functions and the Introduction to Professor Clifford's Remains belong however to the last seven years of his life. d Xxvi BIOGRAPHICAL SKETCH. promoted the well-doing of his pupils, and what his views were about the reforms desirable in mathematical teaching. The present volumes show what splendid contributions Henry Smith made to science during the short twenty-nine years of his speculative activity. Never- theless, it must, I think, be admitted that his unrivalled powers were often employed upon work that scores of able men might have been found to do efficiently, and which his friends should not have asked him nor he have consented to undertake. From 1850 to 1870 he was Lecturer at Balliol, not being able to afford to give up his Fellowship, and having scruples about retaining it if he did not teach, as the number of Fellowships was limited and the stipend of a Lecturer was too small by itself to remunerate any one for the work. It must be borne in mind that during part of this time he was also Savilian Professor, and during the whole of it he was constantly doing other College and University work*, assisting backward men, or taking part in examinations, or serving on Univer- sity Boards and Committees. In 1873 he freed himself from the worst of this drudgery, the College Lectureship, by accepting a flattering and generous offer from Corpus Christi College of a Fellowship upon that foundation f. Not long afterwards he obtained the Keepership of the University Museum, left vacant by the death of Professor Phillips. The office gave him a pleasant house, a small stipend, and not very uncongenial duties, half as master, half as servant, * The ^faster and Fellows of Balliol College, for instance, once asked him to give a course of lectures on the Schoolmen ; and he complied. + A friend at Balliol writes : — ' "We knew perhdps better than others how necessary this relief was to Henry Smith, and we rejoiced that it had come to him ; we knew likewise the perfect loyalty towards his old CoUefte which prompted his resignation. Nevertheless, it was a grievous thing to us that he should be obliged to leave our body. Never had we felt so bitterly the difference between a poor foundation and a rich one. Henry Smith, as Steward of Common Room, was our chairman on social occasions, especially at our annual " gaudy " on St. Catharine's Day. The last speech he made in this capacity was immediately before his migration to Corpus. He assumed a playful tone, and tried to amuse us by various quaint comparisons into forgetting our loss, but he was quite unable to subdue his own emotion, and he was weeping himself before he had made us laugh. This was the only time that I ever knew liim break down. Though ceasing to be a Fellow, he continued to give us the benefit of his presence and counsel at our College meetings. By the next St. Catharine's Day the keen and constant interest which he took in our affairs had somewhat reconciled us to the change, and this feeling was warmly expressed by the Master in proposing Henry Smith's health. I remember the Master's con- cluding words, which struck me at the time as a note of warning, and which have now a sadder •ignificance : " I will only venture to express the hope that he will not suffer himself to be numbered among those men of varied powers and charming manners who have given up to society and business what was meant for science and posterity." ' BIOGEAPHICAL SKETCH. XXVU which sate lightly upon one who was genial and full of instinctive tact. Never- theless, it cannot be said that his work was sensibly lightened for any long time. Partly, he was himself to blame. He had a speculative element in his nature, and had invested so much money in mines — almost always, I am afraid, unremunerative — that it became important now and again to eke out his regular income. I remonstrated with him very strongly, when he added the duties of Mathematical Examiner at the University of London to his other heavy work (1870), for he seemed to be breaking down at the time he undertook it, and I felt sure that whatever he did for the day's need was so much taken from more enduring labours. It seemed however as if the world was in a con- spiracy to force duties of every kind upon one whose talent was so flexible and whom men of all opinions agreed to welcome as a coadjutor. He was for years a member of the Eoyal Commission on Scientific Education, having been appointed in 1870, and he drafted a large portion of its report. In 1877 he became a member of the Oxford University Commission under Lord Salisbury's Act ; and in the same year he agreed to be chairman of the new Meteorological Office, the governing body of which was practically nominated by the Royal Society. This latter work was specially congenial, and the associates were so considerate and able as to give a charm to toil ; and Henry Smith enjoyed the fortnightly visit to London, and the temporary rest from the turmoil of Oxford business. Still, when all is said, it can hardly be doubted that the labours of all these various oflfices meant a partial interruption of nobler toil and may have hastened a premature death. It may perhaps be said, and not without some truth, that those who knew of the condition of his health should have refrained from heaping work upon him and should even have compelled him to take a long term of real rest. But in fact these demands came on him from several and distinct quarters, and what might seem to each person or group of persons making the demand a light and congenial undertaking for the always gracious counsellor of ' golden speech,' became, when added to the aggregate of such undertakings, a serious, perhaps even a fatal burden. The truth is that his presence was always welcome on Boards and Committees ; for he possessed the rare gift of suggesting some middle course which would often efiect an agreement between persons who had been advocating opposite points of view, and of so bringing about a welcome end to a weary discussion. d 2 XXviii BIOGRAPHICAL SKETCH. It will be a melancholy satisfaction to some of the friends with whom Henry Smith went on holiday expeditions, to reflect that these intervals of rest were not t)nly periods of unmixed enjoyment to him, but probably helped to prolong his life. The letters he wrote from Greece in 1871 will not bear reproduction, but are full of the pleasure he experienced. ' We certainly had a most perfect voyage.' ' Except Pylos, I don't think we missed seeing anything we could have seen. Grant Dufi", in a place he has not seen and wants to see, is quite perfect ; and we shall now work morning and night, till we have done Athens well.' 'All that is in Athens we have done to the greatest perfection,' is the comment in another letter. If only excursions of this kind could have been more frequent! The last I find commemorated is a visit to Rome in 1879, when Henry Smith represented the Meteorological Council at the International Meteorological Congress. In 1878 Mr. Gathome Hardy, who then represented the University of Oxford, was raised to the Peerage ; and the Oxford Liberals determined to bring forward Henry Smith for the vacant seat, in the hope that his great personal popularity and unrivalled academical reputation might win over many votes from moderate Conservatives. Moreover Henry Smith was not emphatically opposed to the Jingo or war policy of the Beaconsfield Ministry ; the test by which Conservatives especially weighed politicians in that particular year. He did not expect success, and he hardly desired it, but he would not shrink from a fight if he was asked to stand forward as the representative of a principle. I am told his friends were sanguine of success for a time. Friends are bound to be ; but no sane looker-on could have anticipated any other result than that which actually took place, that the Conservative candidate would be elected by an overwhelming majority. I have never felt that, in this jmrticular instance, the rejection of an eminently good and wise man was uncon- ditionally to be regretted. Personally, I have no sympathy with the doctrine that scholars are out of place in Parliament — a doctrine which would have excluded Macaulay, Gladstone, CornewaU Lewis, Goschen, Fawcett, Grant Duff, Morley, and Bryce among office-bearers of recognised ability, as well as Grote and Mill and a host of others who have added distinction to tho House of Com- mons although they never attained to office. I am convinced that Henry Smith would have been as popular in the House of Commons as he was everywhere else, would always have been listened to when he spoke, and would have spoken BIOGRAPHICAL SKETCH. XXIX with effect. Still I cannot persuade myself that his magnificent powers could have been adequately employed in debating or administering : and I am certain that dozens of inferior men would have played their part as usefully in St. Stephen's ; while there was no one but himself in England so peculiarly fitted to increase knowledge in one very difficult and abstruse department of enquiry. If however Henry Smith had ever gone into Parliament, he would have been something more than a representative of learning or even of academic Liberalism. He had a genuine sympathy with the poor of his own land ; and his last public appearance anywhere was in the Oxford Town Hall to support a resolution by Mr. Arch in favour of giving the franchise to the agricultural labourer. The speech then delivered wUl bear reproduction for its own merits, and is a good specimen of the speaker's style ; that of a man thinking aloud in simple words, yet with an instinctive perception of rhetorical effect. Professor Henry Smith said it was as a Liberal that he would say a few words with reference to the resolution before them. In the course of his Ufe he had found himself sometimes on the extreme left of the Liberal party, sometimes verging towards its right — for every party had a right and left — and pretty often about in the middle of it. He was bound to say that his belief was, in the first place, that the whole of the Liberal party, right, left, and middle, was unanimous in thinking that the National Agricultural Labourers' Union had rendered a great service to the United Kingdom. He further believed that the whole of the Liberal party rejoiced to think that the gi-eat benefit which that Union had conferred upon the agricultural labourers of this country would remain for ever associated with the name of Mr. Joseph Arch. He would endeavour to support the resolution by an argument different a little from those they had heard. The extension of the household franchise to the counties was inevitable ; whether they liked it or not, it was a thing which must be done. He believed there were but few men in this country — sensible men, men who looked at what was around them, and who listened to what was said — but felt that it was inevitable. He was one of those who, when it was clear that a thing must be done, believed that the sooner it was done the better ; and if it were for that reason only, he would heartily support the resolution. But, in addition to that, he did believe that the extension of the franchise to the great classes who now were excluded from it would, as had been well put before them already, exercise a beneficial influence upon the future course of their legislation. It might be true that some persons might ask what would the agricultural labourer and the rural artisan do with the franchise when they got it ? They would do like other people. He feared they would do some mischief, for he knew no class of his countrymen among whom there were not some who did mischief with any right that was entrusted to them, but he firmly believed on the whole they would exercise the franchise XXX BIOGRAPHICAL SKETCH, for good If he were told they would exercise the franchise for selfish objects, for objects peculiar to their own class, he would say let it be so ; but if so, what would they say to the other classes who already possessed political powers? Could any one of them send a representative who could say that his hands were free from selfish legislation? If they must count — for, alas, they must— upon some strain of selfishness in their common nature, at least let them take care that their representation was not one-sided, but that at any rate each class had a fair means given it of defending itself from others. It was for these reasons that he for one most heartily supported the resolution. He hoped to see the household franchise extended as rapidly as possible to the counties, and he was not one of those who shrank from a still greater consequence which would come when that great measure of enfranchisement should be followed by an equally sweeping measure of redis- tribution of seats. From that platform Henry Smith went home to die. Overwork and sedentary work had gi-udually undermined his constitution. When I last saw him in 1879 he still looked well, and in some respects, from having filled out a little, less delicate than as a young man, but I noticed that he was less capable of sustained effort. In 1881 premonitory signs of a break-up of the constitution showed themselves, and were unhappily not heeded as they should have been. First he suffered from his digestion, and had to put himself under Sir H. Thompson's care ; and then a stoppage in one of the veins of his leg confined him for many months to the sofa, and made all but occasional carriage exercise impossible. He seemed to be tiding over this illness, when a rush of University work threw him back again into the condition of an invalid. When he spoke at the Town Hall meeting he was suffering from a cold. The exposure and excitement were followed by congestion of the liver, which was the more dangerous after the severe attacks which had followed on the Roman fever, from which he suffered in 1845. On the morning of Thursday (February 8) there seemed to be a change for the better, but at noon the worst symptoms returned ; and Sir William Gull, who had been telegraphed for and who arrived about eight o'clock, held out little hope. About four o'clock next morning (Friday, February 9, 1883), the patient's state was declared desperate, and three hours later he passed painlessly away. He was buried (writes a friend who was present) at St. Sepulchre's Cemetery in Oxford on Tuesday, February 13. So great a concourse of undergraduates as well of senior members of the University and friends and strangers from a distance has rarely been seen on an occasion of the kind in an English BIOGRAPHICAL SKETCH. XXXI University. An academic funeral is always an impressive spectacle, and the long line of the procession and the scene in St. Paul's Church and around the grave in the cemetery will never be forgotten by those who were present. For once there was no discordant criticism over a grave. Not only did all agree to speak with tenderness and admiration of the dead man, but there was a singular consent of opinion as to his character and pre-eminent intellect. The funeral procession that carried him to his resting-place was nearly a quarter of a mile in length, and included every man of position or note in Oxford and many others distinguished in their various ways who had come from every part of England to pay the last honours to a dead friend. The Times wrote of him as 'one of the most remarkable men of his day,' and the Spectator declared that ' it would be difficult among the world's celebrities to find one who in gifts and nature was his superior.' ' Some of us,' said Professor Stubbs (now Bishop of Oxford), in a University sermon, ' can remember the youth of brilliant promise, of almost unparalleled achievement in all our studies ; all of us have before our eyes the manhood of indefatigable energy, of most generous devotion, of most kindly and effective sympathy with all good work ; the entire expenditure of consummate accomplishments and of every bright gift on the work of Oxford.' Perhaps however no words were more frequently before men's eyes or in their thoughts in connection with Henry Smith's death, than a tribute which Sir M. E. Grant Duff had once paid him in the House of Commons, in commenting on his nomination as one of the Oxford University Commissioners. He said : — 'Professor Henry Smith is not merely in the first rank of European mathematicians, but he would be a man of very extraordinary attainments even if j-ou could abstract from him the whole of his mathematical attainments. He was the most distinguished scholar of his day at Oxford. . . . But Professor Smith's extraordinary attainments are the least of his recommendations for the office of Commissioner. His chief recommendations for that office are the solidity of his judgment, his great experience of Oxford business, his services on the Science Commission, and his conciliatory character, which has made him perhaps the only man in Oxford who is without an enemy, sharp as are the contentions of that very divided seat of learning.' To myself, who am no mathematician, and who therefore cannot estimate Henry Smith's intellectual power in the departments where it was highest, it has seemed also, as it seems to Sir M. E. Grant Dufl', that I have never known his equal or perhaps one who could be classed with him. What always im- Xxxii BIOGRAPHICAL SKETCH. pressed me however was not so much his marvellous versatility or his thorough mastery of everything he touched, or his conversational brilliancy — though none of all these can be separated from my recollections of him — as his singularly clear judgment, combining insight into the essential truth of whatever he examined and balance in the summing up of it. Never did genius more completely take the form of sublimated common sense ; and this effect was undoubtedly enhanced by his unassuming manner. What he had to say was never thrown into a doctrinaire form, half dogma, half epigram, but was stated in the simplest possible words. Sometimes no doubt an opinion given in this way would attract less attention than it deserved and it would certainly be less effective than a brilliant paradox. Gradually but surely those who met Henry Smith, or who came to him for counsel, perceived that his insight was unerring, and learnt to defer to his judgment, the less reluctantly as ' he had the great art of never pressing a victory home, and of bearing defeat with pleasant equanimity *.' Perhaps it was this faculty of judgment which kept him from being over- weighted with his learning. He had read many books which even scholars rarely open, and he never forgot what he had once read. I remember for in- stance how he gave me on one occasion a most amusing account of the Letters of Synesius, which Kingsley's Hypatia had, I think, induced him to look up. His knowledge of Protestant Hymnology was curiously intimate and wide : and, when he assisted a friend to compile the University Hymn-book, his recitals from memory of whole hymns by Wesley and others impressed those who were present as very remarkable. Even his private friends, however, only learnt by rare glimpses what his acquirements were ; and in general society, though he never affected to be other than a scholar, he impressed those who met him as a man of the world with perhaps unusual cultivation. His friends sometimes compared him to Pascal, with whom he had many points of resemblance, the combination of mathematical and general ability, a keen wit, an extreme reserve, and an unfortunate lack of personal ambition. There was, however, one remarkable difference. Pascal, who has recorded the oi)inion, ' Diseur de bons mots mauvais caract^re,' meaning, I suppose, that an epigram is a truth pared to a point and twisted into a barb, was yet seduced by his genius into endowing the world with a book that scathed and blasted the * Spectator, Feb. 17, 1883. BIOGRAPHICAL SKETCH. XXXlll cause and the men he assailed. Henry Smith's tenderness of feeling interfered with his command of literary form. He had a feminine instinct for avoiding what- ever would give pain, and never allowed his buoyant spirits to betray him into a word that might seem harsh, or his inimitable persiflage to pass the boundary line into sarcasm. Those who heard him talk were conscious of wit that played round every subject with a perpetual sparkle, and that left a delicate aroma behind ; but no one ever knew it employed as a weapon of offence. Reading over his private letters I find the same kindliness, the nearest approach to personal satire being perhaps the description of a heavy dinner, 'with four pieces de resistance, not including X and Z,' two rather overwhelming talkers. Therefore if Henry Smith had ever written on any of the subjects on which he felt strongly — and he was an ardent Liberal on every University question and on almost every political topic of interest — I cannot doubt that he would have adopted a style of earnest simplicity, and would have trusted for effect to argument, enhanced at most by a restrained eloquence. Bearing in mind that he was confessedly one of the most brilliant talkers of his day, so that every obituary notice dwelt lingeringly upon this trait, and considering how easily the playful but keen humour might have been transformed into caustic satire, I can only wonder at the mixture of kindliness with strong self- discipline that prevented even an occasional lapse. Both in this matter and in his judgments of men and things, a singularly fine character gave law to the intellect. He was clear and just in expression because he was accurate and truthful in thought ; he was irreproachable in speech, because he never allowed himself to cherish an ill-natured thought. Any one who has been often in the society of brilliant talkers can fiardly have failed to notice how little of the best conversation is of a kind to bear record or is practically remembered. Dr. Johnson was singular in attracting an unrivalled biographer, who took notes unblushingly, and was skilful enough in literary form to polish up what he took ; and Sydney Smith's fertility was so great that some of his mirth has survived him : but of George Selwyn and Luttrell, of Fox and Canning, of Macaulay and Bagehot, we know disappointingly little. Two or three trifling instances may serve to show what Henry Smith's manner was. He was once winding up a mathematical lecture by explaining a new solution of an old problem. ' It is the peculiar beauty of this method, gentlemen,' he concluded, 'and one which endears it to the really scientific e XXxiv BIOGRAPHICAL SKETCH. mind, that under no circumstances can it be of the smallest possible utility.' He was obliged to pass through France in 1870, when fortune had just turned against the French armies, and the cry of treacheiy was raised everywhere. A guard noticed the tall Englishman with a blonde beai'd and spectacles, and instantly denounced him as a German spy. A suspicious crowd collected in a moment around the carriage. 'Gentlemen,' said Henry Smith with an amused smile, * I speak French very badly, but not I hope with a German accent.* The proof and the speaker's impressive serenity cairied conviction, and the crowd melted away. ' You take tea in the morning,' was the remark with which he once greeted a friend, ' If I did that I should be awake all day.' A friend mentioned to him the enigmatical motto of Marischal College. ' They say; what say they; let them say.' 'Ah,' said Henry Smith, 'it expresses the three stages of an undergraduate's career. In his first year he is reverent, and accepts everything he is told as inspu-ed : " they say ; " in his second year he is sceptical and asks " what say they ? " and " let them say" expresses the contemptuous attitude of his third year.' At a time when English society was perhaps extravagantly fluttered by Lord Beaconsfield's apparent success at the Berlin Conference, Henry Smith reduced the event to something like its proper proportions. ' Dizzy,' he said, ' has taken John Bull to Cremome, and the old gentleman is rather pleased to have been there.' On the news that a distinguished friend, who was also markedly pessimist by temperament, had been appointed to a high post in India : ' How fortunate ! ' was the remark ; • it will give him another world to despair of.' He summed up X, a brilliant writer but inconsecutive thinker, in the criticism, ' X is never right and never wrong ; he is never to the point.' It is sometimes said of loveable men, that they difiuse their affections BO evenly as to be incapable of strong personal attachments. With Henry- Smith to be a friend once was to be a friend for life. The masterly biographical sketch which he wrote as an introduction to Conington's Miscellaneous Writings will give a measure of one friendship that lasted from school days till it was interrupted by death. Professor RoUeston, who could hardly ever speak of him without some epithet such as ' the golden-mouthed,' confided iiis family when he died to Henry Smith's care, and the trust was accepted and discharged with exemplary fidelity. Probably no other great student was ever so ready its he always was to put aside books and papers when a friend entered the house. I BIOGRAPHICAL SKETCH. XXXV Yet his nature, genial and hospitable in the extreme, was not what is called an,- effiisive one. He has noticed in the life of Professor Conington, that that great scholar, after he underwent a spiritual conversion, used to speak of his experi- ences unreservedly though in the simplest language. On this as on every subject of delicacy, Hemy Smith was absolutely reticent. He would discuss religious topics if they were started as matter of interest, but I never knew him talk of his own faith, and I should be slow to believe that he ever did. My impression is that he accepted Christianity not only as a habit and a conviction, but as a rule of life, and in fact his character can scarcely be explained, except by ranking him with those who feel that they are ' ever in the great Task- master's eye.' I think he regarded much popular theology as irrational, and much fashionable doubt as a mere winnowing of chaff. Some of the weightiest words I ever heard from him were on religion. Beyond this I can say and surmise nothing, and his letters are not more unguarded than his speech was'\ The one question as to Henry Smith's character that appears to be still un- decided is, whether his inaptitude for self-assertion, his scorn of personal ambition, his severe acceptance of duty in whatever shape it came to him, are to be regarded as blemishes or excellences. The distinguished friend who wrote about him in the Spectator] has shown in thoughtful and wise words how much there is admirable in the 'philosophic life' — 'life of exemplary moderation, far removed from even a suspicion of worldliness and vanity.' ' Great moral gifts,' as the writer goes on to say, ' can be found when occasion demands them ; talents grow on every tree. But the serenity of heart which enables its possessor to wear the gifts of genius with sobriety, and to use them nobly and well, without seeking to expend them in the purchase of fame, or wealth, or of advancement, is a quality which modern society little cultivates and seldom sees.' It may seem to those who ponder this temperate and lofty apology, that it is a sufficient answer to the regrets I have freely expressed in this sketch over genius that was often lavish of itself on work for the moment's need or work of ordinary compass. Let me say for myself and for those who think with me, that we never desired wealth for Henry Smith except in such measure as might free him from sordid * He on more than one occasion spoke to me on these subjects. His position was perhaps most simply expressed in a conversation in the course of which I remember his saying that the essential features of the Cliristianity lield to-day were held in the time of Justin Martyr.— N. S. M. t The ai-ticle is printed at p. xlvi. e 2 Xxxvi BIOGRAPHICAL SKETCH. necessities, or fame except as a recognition of what he might achieve ; and that the world's opinion or the state's honours could not have raised him in the estimation of those to whom he waa already above all men. Our feeling about him was essentially what Newton expressed when he said that ' if Cotes had lived, we should have known something.' It is very possible that those who saw how much of his time Henry Smith gave to Examinations, and Boards, and Commissions, and who unconsciously estimated the range of human effort by their own measure, did injustice to the special capacity I have noted in him for carrying on consecutive work in stray moments. It has been said by one who can speak with authority on such a subject that Henry Smith was ' the author of mental achievements in the most abstract and complicated of the sciences, which will rank as scarcely second to any in the century.' On this matter the collected works will be conclusive evidence. What, however, surviving friends will and must feel is that genius is as rigidly bound to husband its powers as mere capacity, and that nothing can be spared from the supreme work of life without loss *. Prove to us that Henry Smith's work was indeed scarcely second to any in the century, and we are constrained to assume that, had his energies been more severely economised, it must have been second to none. Certainly the ordering of these things is not in our hands. He who gave the perfect intellect gave also the fine temperament, the tenderness that shrank from disobliging, the modesty that esteemed no duty undignified, the absolute disregard of self To us who knew him, let me repeat, the man was always greater than any possible work he might do, though we set no limits to its possibilities ; and to us the ever-green wound of his loss is partially compensated by the remembrance of an ideal character. What we grieve for is that generations that did not know him as we have known him will try him by the only standard possible, that of his completed work, and will give him less than the measure of his real capacity, though they can never refuse to number him among the great names of the century. ' * I may add, however, to this that, frequently as I urged on H. Smith to turn a deaf ear to some of the too many supplicants for his time, and to give up some of his less important occupations, his answer to me always was that lie did ' get all that he could out o£ himself as it was ; that in truth his greatest work could only be done now and then, and could not be reeled off the mind indefinitely. Much interval was necessary to him. — N. S. M. RECOLLECTIONS OF PROFESSOR SMITH. I BY PEOFESSOR JOWETT *. MY recollections of Professor H. J. S. Smith extend over about 40 years. I first heard his name mentioned in the year 1843, by the late Archbishop of Canterbury, then recently elected Head Master of Rugby, who told me that there was a boy in the School quite deserving of a place by the side of Conington and Walrond, who were the great names of Rugby in those days. He was going to try for the BaUiol Scholarship. At the end of the year, on that occasion, he beat all the other candidates, of whom one was the late Sir Alexander Grant, elected Scholar at the same time with him. I remember him in the viva voce part of the examination, a youth of eighteen, rather overgrown and stiff, as youths of eighteen are apt to be, construing Latin and Greek authors in a pious and evangelical tone of voice, which provoked a smUe in the Examiners, but with never-failing accuracy. The old Master, as we used to call him, took up his English Essay and showed it to me, saying, in his emphatic way, ' There 's mind in that.' The subject given for Latin Hexameters at that examination was the Pelasgi, of whom he did not forget to mention in his verses that 'they worshipped nameless gods.' Meeting Arthur Stanley on the Woodstock Road the day after the election, he congratulated me on our having chosen a youth whose fame had preceded him at Oxford. He more than justified the promise which he had given. Though not a poet or creative genius, he was, I think, possessed of greater natural abilities than any one else whom I have known at Oxford. He had the clearest and most lucid mind, and a natural experience of the world and of human character hardly ever to be found in one so young. He took up all subjects at the right end ; he knew whereabouts the truth lay even when he was imperfectly ac- quainted with the facts. And he was the most amiable and good-natured of * Professor Jowett's death has deprived these ' KecoUections ' of the author's final revision. XXXviii RECOLLECTIONS OF PROFESSOR SMITH young men. I might apply to him the words in which Plato describes the youthful Athenian Mathematician, Theaetetus, where he says : ' In all my ac<|uaintance, which is very large, I never knew any one who was his equal in natural gifla He had a quickness of apprehension which was almost unrivalled, and he was exceedingly gentle. There was a union of qualities in him which I have never seen in any other, and should scarcely have thought possible, for quick wits have generally quick tempers . . . but he moved surely and smoothly and successfully in the paths of knowledge and enquiry. He flowed on silently like a river of oil. At his age it was wonderful. He was also surprisingly liberal about money, though his fortune was only moderate' (Thesetetus, 144). The facility with which the youthful Scholar of Balliol picked up all sorts of knowledge was equally wonderful. During the first two years after his election to the Scholarship he was struck down by a serious and almost fatal illness, and did not come up to Oxford until what is usually the third year of residence had commenced. In the interval, residing in Italy he acquired a con- siderable knowledge of Roman inscriptions and antiquities, and also of modern languages. Within the year and a half which remained of his Undergraduate course he obtained the Ireland Classical Scholarship, and a double fii-st. Two years later, the Senior Mathematical Scholarship was awax'ded to him. Similar honours have only been gained by one other person — the late Very Rev. G. H. S. Johnson, Dean of Wells, an eminent man, but little knowai, who, from ill-health, was unable to do justice to his great natural talents. After he had taken his degree, he at one time thought of going to the Bar, for which he was very well suited — he would have risen rapidly to the high places of the profession. But the feebleness of his constitution when a young man led him to abandon this intention, and he soon settled down in the post of Mathematical Lecturer at Balliol College. The other tutors were the late Rev. E. C. WooUcombe, Dr. Lake, the present Dean of Durham, and myself. The Bishop of London (Temple) who preceded him in the office had just left us. In those days he was almost equally a lover of Classics and Mathematics. There was a time when he was quite divided in his allegiance between them, and used to say, in his free and easy way, that he ' must toss up a shilling to decide.' Even in the last years of his life he was in the habit of taking with him Greek books to read during the Vacation. In conversation he left tae impression of being a well-read scholar, and a real critic, who was never led away by ingenious conjectures or imcertain fancies. For some time he was intending to edit the Timseus of Plato for the Clarendon Press, but he never had leisure to carry out BY PEOFESSOR JOWETT. XXXIX this project. He finally determined, and probably he was right, to make Mathematics the chief work of his life. The Mathematician is more cut off by his pursuits from his feUow-men than the student of any other branch of knowledge. He has interests which are locked up in his own breast, pleasures and also pains which he cannot communi- cate to others; the better part of him is moving about in a world of numbers and figures which have no connection with ordinary life (cp. Plato, Thesetetus, 1 84 d). His study is apt to become a passion with him and affects his character. I am sure that this was true of Professor Henry Smith. It was the smaller part of him which we knew or could appreciate. His mathematical speculations could have been shared by a very few, not more than two or three, of his contempora- ries at Oxford. Yet he did not withdraw himself from business or society. He was not the silent philosopher who is lost in reverie, or who, while acknowledged to be a mathematical genius, is pointed at by mankind as a poor and eccentric mortal. He was a thorough man of the world and greatly liked by everybody. He was very manly in his bearing, and quite free from shyness and nervousness in any company. He had a kind greeting for servants, and felt a real kindness for them — they were devoted to him. His manner of behaviour towards all sorts and conditions of men might be described as exhibiting a singular 'urbanity.' He was decidedly good-looking, and there was a certain intellectual distinction in his features and expression. It is necessary to combine these various aspects of him if we would duly estimate him. He was everywhere, and known to every- one, the life and soul of a social gathering. But he was also a thorough student, and an omnivorous reader, passing several hours of the day in abstruse Mathe- matics, but nevertheless acquainted with all new books, and on a level with every recent scientific enquiry. He went on teaching at Balliol College as Mathematical Tutor for about thirteen years ; at the end of that time he was appointed Professor of Geometry ; he then combined the duties of Tutor and Professor. While only a Tutor of Balliol he had hardly any pupils worthy of him. The College, having at that time no Mathematical Scholarships, had seldom any good Mathematical students (those who were being usually men who read for double honours). His duties were, for the most part, confined to the preparing and examining men for Responsions. But he never thought it beneath him to take pains with any one, and he was an admirable teacher. He used to have his pupils on a Sunday afternoon to be examined by him, and would tell them that 'it was lawful on the Sabbath Day to pull an ass out of the ditch.' The better men were of Xl BECOLLECTIONS OF PR0FES8OE SMITH opinion that they learned more from him in a few minutes than from another in a whole hour. He was constitutionally apt to be late and irregular in lecture, and on occasions of business as well as at a dinner party was often the last to arrive, but every one was very willing to wait for him. The circumstances of the University hardly admitted of his raising up a School of Mathematical pupils, but he was the life and centre of the study while he was with us. He was very desirous to promote the interests of Natural Science in Oxford, and was in favour of some measure which would have made the knowledge of a portion of some one of the Natural Sciences the condition of obtaining a degree. The teachers of these sciences had long been fighting a battle against the older traditions of the University ; they had now become the study of a few, but he clearly saw that they could never truly flourish until an interest in them was more generally diffused and they had a congenial atmosphere. But he was also the best friend that the older studies then had in the University, for he could speak with authority, and he was firmly convinced that in education Science should not supersede Literature. He deplored equally the want of literary culture which he observed in many scientific men, and the gross ignorance of the most general facts of Science which prevails in the world at large, especially at English Uni- versities and Public Schools. In a similar spirit he was anxious to encourage at Oxford the study of Medicine and also of Engineering, thinking that they would supply a missing link between the Physical Sciences and the older studies of the University. A considerable portion of his time was devoted to College and University business. Though he transferred his name to C. C. C. about ten years before his death and nominally ceased to be a member of Balliol College, he con- tinued to show the same earnest interest in its concerns which he had always done. He took the same part in its Examinations and College Meetings — the only difference being that he no longer received the stipend of a Fellowship from it. There was never any one more affectionately regarded by the Fellows, or whose opinion had greater weight with them. He had the art not only of doing business well, but of making it pleasant, often with a slight jest or play of words smoothing away difficulties. I do not remember his ever having had a quarrel or difference with any one in the University. It will be easily understood that such a man was well adapted to keep men together and to carry things forward. At the Hebdomadal Council, where he usually appeared rather late in the day, he gave life and animation to every discussion. He seemed to say things in a better way than anybody else, and in an argument there was no one who was a match k BY PROFESSOR JOWETT. xli for him. When a new measure had been put into form by the Council he was very often selected to carry it through Convocation, his popularity and his manner of speaking having great weight in that assembly ; and it was whispered that ' the Council relied for the success of their measures too much upon Henry Smith's oratory.' Though well aware that the order and discipline of the Uni- versity must be maintained, he was always a very earnest supporter of freedom, and a great enemy to the imposition of useless restrictions upon Undergraduates. He was indulgent to the failings of young men, and felt a humane pity for persons who had lost their character. He was one of whom it might be said that ' he would have stood by a friend, not only in adversity, but in disgrace.' Two occasions on which he distinguished himself were long remembered by those who heard him, — once in the Common Room, more than thirty years ago, when some of the elder members of the College sought to impose a new-fangled test upon the undergraduates instead of the time-honoured Thirty-Nine Articles. He pleaded earnestly for the retention of the latter, alleging that ' old chains were smoother and easier to the wearer of them.' The other occasion was in Congregation, about twelve years ago [1880], when he introduced a measure granting privileges to Colonial Universities, and drew a sketch of the growth of the London University, and of the mistaken policy of Oxford and Cambridge in their opposition to it. He was not an orator, but a very good speaker, who had the faculty of thinking when on his legs, never faltering for a word, able to strike out, right and left, good-humoured and telling blows. His speeches were clear and luminous, and they also had the merit of keeping up the attention. Above all, he had tact. He said what he ought to have said, and abstained from giving needless offence. As a writer, he never attained to considerable eminence. He was the author, when quite a young man, of a very clever review in the Oxford Essays of Sir David Brewster's ' More Worlds than One.' In this paper the fallibility, both of men of science and of theologians, was impartially exposed, and I re- member that Bishop Temple remarked, after reading it, that ' the author could do many things well, but that he would write better than he did anything else.' The prophecy was not destined to be fulfilled. His mind was drawn in another direction, and he had not the poetical gifts which seem to be indis- pensable in a great writer, whether of poetry or prose. He was wanting in initiative. Though a very able supporter of the plans of others, he rarely, if ever, took the first step in introducing a measure himself. He was easy-going, not burnt up with a fiery zeal for change, but satisfied in f xlii RECOLLECTIONS OF PROFESSOR SMITH general with the world as it is, and really, I think, somewhat deficient in prac- tical originality. He was contented to be a follower rather than a leader in most of our University contests. When he was brought forward a few years ago, rather against his will, as a Candidate for the representation of the University in Parliament, he was, I believe, absolutely indifferent to the result. I never knew a man, possessing so much ability, with so little ambition. Hence he inter- fered with no one, and as no suspicions were entertained of him, everybody was willing to do justice to his great merits. Nor had he any sympathy with new opinions in politics or theology. In politics he would have professed himself a Liberal, but he was not an advanced one. He was willing to talk about them, and his views were always worth hearing, because they were not strained out of newspapers, but the result of his own reflections. He was an acute political economist, a disciple of Mill and Ricardo, not much interested in the wider field which is sometimes claimed for the science. Nor was he at all disposed to under- value the influence of theology. He was well acquainted with the results of German criticism on the Scriptures, but they seemed to make no difference to him. When he first came up to the University he was an Evangelical, and, for a while, retained his old belief Indeed, some years after, on the occasion of a high-church sermon at St. Mary's, he would say, with indignation, ' that was not the sort of religion in which he had been brought up.' But, in time, the old clothes of his youth naturally fell off — he had out-grown them, and there remained a blameless character, a singular kindness and generosity, a love of justice and fairness, and a sense of religion which was wrapped in im- penetrable silence — it was one of the subjects of which he least desired to talk. He was very reserved. Like many other persons who pour themselves out fi'eely in conversation, there was the appearance of abandon, but there were many subjects about which he rarely, if ever, spoke. One of these was himself He was probably the confidant of many, for no man could give better advice in a difficulty, or was more willing to assist others. There was a feeling that he could be absolutely trusted, and even if a foolish thing were said to him, that he would not repeat it. His insight into human character was said, by one of his friends, to be ' terrible,' but it was never used by him except for some kind purpose. He could see through the vanity and folly of a friend, and yet retain a never- changing affection for him. Of his own life, he seldom or never spoke ; he was not an egotist, and his own sayings or doings did not seem to interest him afterwards. It is difficult to give an idea of his conversation. It was gay rather than BY PROFESSOR JOWETT. xliil serious, full of life and chaff, arising naturally out of the circumstances of the hour. If a stranger had come across him in a railway train, or had been his companion on a voyage, he would probably have found that this unknown person was one of the most agreeable men he had ever met. It was a great pleasure to have a t6te-k-t6te with him, for he was not one of those who required a company in which to show off. I have often decoyed him into my room for the sake of having a chat with him, and when once there, he was very willing to stay, for he was one of those who like to have a talk out and did not hurry away when the clock struck. In society he was ready to talk to every one, and every one was ready to talk to him. He had the art of setting people at their ease. He would at times break out into fits of laughter and joviality, which showed that the original Irish nature was not extinguished, but only kept under by him. Stories were repeated of his performances at Meetings of the British Association, which must greatly have enlivened that sedate assembly. He was certainly a wit, but his good sayings were of too delicate a fibre to be transplanted. To use BosweU's expression, his bons mots ' would not carry.' But they were the delight and admiration of those who heard them at the time. They possessed also one of the highest qualities of wit and humour, spontaneity. They were made on the spur of the moment with reference to something which was said or done at the time. And this very quality tended to impair their effect on those who were not present when they were first uttered, and did not know the occasion which had given rise to them. A hght irony seemed to be always playing about his mind. It was the form under which he inclined to regard all human things, for he was very unimpassioned. An old school-friend would sometimes be the target at which he aimed. The gi-eat scholar, Professor Conington, a man so unlike himself, that their mutual friends wondered what could be the tie which united them, was often the butt of his humour. But the slight humiliation to which he was subjected was more than made up to him by the constancy and faithful attachment of his friend, who afterwards collected his literary remains and wrote his life. He was a little provoking to some others, especially to those who were too earnest or of too pushing a temper. He knew how, in Aristotle's language, 'to overcome seriousness by laughter,' or in other words, to make such persons appear slightly ridiculous. An enthusiastic friend might have thought him deficient in sympathy, but he was really always kind and considerate. I hardly venture to repeat some of his good sayings, lest, detached from their surroundings, they should seem not to justify the high opinion which has been expressed of his conversational gifts. They are not of course of the quality of f 2 Xliv RECOLLECTIONS OF PROFESSOR SMITH the best sayings of Charles Lamb or Sydney Smith, yet they are such as might have been said by them. The reader is requested to bear in mind their im- promptu or occasional character. He who made them could have made many such every day of his life, and never aspired to be a wit, but only to amuse himself and his companions. At any rate they may serve to remind his friends of pleasant hours which they passed with him, never to return. A friend told him of a rather ponderous jest made by Sir George Lewis, who, when Minister of War, once proclaimed in the House of Commons in a loud voice that he had ordered experiments to be tried respecting the comparative effect of ' short and long bores.' To this heavy piece of artillery Henry Smith iixstantly replied by asking whether he was not aware that ' smooth bores ' were the most deadly of all. — Another friend said to him : ' What a wonderful man Ruskin is, but he has a bee in his bonnet.' • Yes,' replied Henry Smith, ' a whole hive of them ; but how pleasant it is to hear the humming!' — The Lectures of a certain College Tutor were reported to be ' cut and dried.' ' Yes,' said Henry Smith, ' dried by the Tutor and cut by the men.' — A dispute arose at an Oxford dinner-table as to the comparative prestige of Bishops and Judges. The argu- ment, as might be expected at a party of Laymen, went in favoiu* of the latter. ' No,' said Henry Smith ; ' for a Judge can only say, " Hang you," but a Bishop can say, " D — n you." ' — The next is of a higher class of wit. Speaking of an eminent scientific man to whom he gave considerable praise, he said : ' Yet he sometimes forgets that he is only the editor and not the author of Nature.' The two remaining ones are autobiographical. He once said to a friend : * C, I was kept in bed by illness when quite young for six weeks ; I then began to study mathematics, and I wish I had remained there ever since.' — Speaking to a newly elected Fellow of a College, he advised him, in the low whisper which we all remember, to write a little and to save a little, adding : ' I have done neither.' These slight jests may perhaps be thought disappointing : it is probable that they are marred in the telling. They were the bubbles which were always rising to the siuface of his mind, and though but poorly reported, may help to give to those who did not know him personally a faint idea of the charm of his character and conversation. Though not rich, he was extremely liberal. He never seemed to think either about gaining or spending. He used to say that not enough money was to be had in Oxford to make it worth while to take trouble about it. Yet a certain love of speculation which was latent in his nature once led him into an BY PROFESSOR JOWETT. xlv unfortunate venture, from which he extricated himself by taking the affairs of a Company into his own hands, and at a considerable loss. For his services as a College Tutor he received a very moderate remuneration, but, having enough for his wants, he never seemed to desire that it should be increased. He did not wish to impose on the College a burden which it could Ul afford to bear. I have endeavoured, in a few pages, to give a sketch of one with whom I was in daily intercourse during thirty-five years of his life, and who I think may be regarded, without exaggeration, as one of the most remarkable persons of his time. Yet he lived and died almost unknown to the world at large. I have sometimes asked myself what was the reason of this contrast between his reputation and his real merits. It has been said that ' the world knows nothing of its greatest men,' but this familiar line, whether true or not, is not the whole account of the matter in his case. The explanation is partly to be sought in his own character. He had no ambition, he had not a strong will, and he had never made himself known to the public. He was once reproached by a friend for 'giving up to society what was meant for mankind,' and the reproof, as far as it appUed to his life at Oxford, was not without foundation. He was not the author of any considerable work. His Mathematical writings, on which his fame chiefly rests, await the judgment of time. Though he managed, in great part, the affairs, not only of the University, but of several other great institutions such as Winchester and Rugby Schools, University College, Bristol, the Univer- sity Commission, the Meteorological Office, the Oxford Museum, of which he was Keeper, and the Ashmolean Society, of which he was the Secretary, he could hardly be said to have left his mark upon any of them, however valuable his services have been to those institutions. To understand his superiority over his contemporaries, it was necessary to have lived with him and known him, to have heard him lecture, to have been with him at a College Meeting, to have enjoyed his society at a dinner-party, or on an excursion of pleasure. He never offended you, never disappointed you, he was never tired or out of humour. His greatness was shown in the peaceful continuity of a private life, not in great actions, or on striking occasions. B. JOWETT. xlvi RECOLLECTIONS OF PROFESSOR SMITH 11. RECOLLECTIONS BY LORD BO WEN*. Great statesmen, successful generals, famous authors, distinguished men of science, eminent theologians — all those who have been raised by industry, talent, or the caprice of fortune, to prominence in a profession — become by degrees actors on whose movements our attention rests, and whose familiar figures are part of the spectacle of life. The public they have interested during their time bids them, when they die, a kindly and sympathetic farewell, retraces their career, counts up their successes, and assesses their general apparent value. Professor Henry Smith, whose loss this week casts a shadow both over Oxford and through many circles of educated men and women, belonged to none of these categories. To by far the greater number of Englishmen, his name is probably unknown. Some will vaguely recollect it as that of a candidate put forward unsuccessfully a few years ago by Oxford Liberals for the representation of the University. Many even of those who are aware that a man in the fulness of his powers is just dead, whose brilliant intellectual attainments have probably not been surpassed by any other of their English contemporaries, may, nevertheless, be surprised at regret so widely felt and so loudly expressed over the loss of one who wrote no great books, patented no great invention, amassed no fortune, made no famous speeches, and led no conspicuous movement, political or social. Measured by the popular measure of publicity and fame. Professor Henry Smith would hardly seem, to most of us, to have been one of the great men of the time. Yet it would be difficult among the world's celebrities to find one who in gifts and nature was his superior. Generally speaking, there is a rough justice in the sentence passed upon intellectual men who achieve no definite worldly success. We surmise, and often with truth, that some weak spot somewhere in their powers has been the cause of their failure to acquire those sublunary distinctions and rewards which coarser and more practical people manage to secure. To the case of Professor Smith, tliis kind of criticism would be inapplicable, for he possessed both the qualities and the character which might have made him famous in many active walks of life. His mental attainments were of the highest order. A finished classical [Reprmted (by permission) from The S])ectator, Feb. 17, 1883.J BY LORD BOWEN. xlvii scholar, a mathematician, in some respects of European distinction, a con- siderable metaphysician, a trained master of most branches of knowledge, literary, economic, and scientific, an adequate linguist, and a man of sound judgment, perfect temper, and wise aptitude for affairs, he combined with his other special excellences a dehcate gaiety of spirit, a brilliant conversational power, which made him one of the most accomplished and attractive ornaments of any educated company in which he moved. To what eminence in public or professional life accomplishments so varied might not have led him, it is difficult to feel sure, if only he had ever plunged into the stream of competition or adventure. But some delicate touch of indifference to worldly success mingled itself with his genius, and he remained to the last content with playing, and with playing well, whatever part fortune brought to him to play. Incessantly occupied in the discharge of duties both of a public and a private kind, that thickened round him as years went by, he was satisfied with what had fallen to his share in the lottery of life, and neither solicited nor ostentatiously avoided anything beyond. The ' note ' of personal ambition seemed absent from his composition. And so it happens that the great public which takes its knowledge of men from newspapers and books, from debates in Parliament and the records of our Law Courts, hardly knew — if, indeed, it knew at all — Professor Henry Smith. As the personal ' note ' was wanting in Smith, so, on the other hand, the intellectual or academic ' note ' was one which he possessed in, perhaps, its most attractive form. Vanity and self-seeking, every form of mental intem- perance and extravagance, seemed to have no place in anything that he ever said or did. The last, the rarest triumph of education, is when it destroys the desire of self-assertion in a man of genius, and substitutes in its place the crowning flower of perfect moderation and equanimity. The greatest of Greek philosophers, in the greatest of moral treatises, has elaborated a theory that virtue consists in a golden mean, and in the avoidance of dangerous extremes ; but when driven into a corner for a standard by which the mean is to be measuied, the illustrious moralist has no better compass to furnish for our guidance than this, — that the golden mean in each case must be that which is defined by the reason of some thoroughly temperate man. The result of Henry Smith's genius and culture combined seemed to make him the very man required by a philosopher for his human measuring-rod. A University life sometimes spoils and sometimes perfects natural capacities, but it usually leaves its mark upon them, whether it be for good or evil. Nobody could doubt but that Henry Smith, as he xlviii RECOLLECTIONS OF PROFESSOR SMITH issued from the Academic mould, was a natural genius, with an impress of his University stamped distinctly upon him ; and Oxford has, perhaps, never had a more happy specimen to produce of her best influence than the late Savilian Professor of Geometry. Smith came from Rugby to the University as a remarkable boy, and won the blue ribbon in all the great intellectual competitions of his undergraduate days. He became in due course a Fellow of Balliol, and joined a Common Room which consisted of a small group of very distinguished men. The present* Master of Balliol was already conspicuous in the society of Balliol Fellows, as the most successful and most energetic tutor of the first of the Oxford Colleges of the period. Among the rest were names of academic fame — Mr. Lake, the present Dean of Durham ; Riddell, an accomplished hero even among Shrewsbury scholars, whose beautiful character and refinement of mind were prematurely lost to the University by an early death ; Archdeacon Palmer, not the least distinguished of a trio of brothers with all of whom Oxford had reason to be content ; Lonsdale, Wall, WooUcombe, Walrond, and a few years later, Newman and Green. These were the days when Oxford, always passing through some phase or other, was entering on a new situation. The Tractarian movement had subsided, but the University was not at rest. A reforming Parliamentary Com- mission was troubling the waters. The old system of close Scholarships and Fellowships was slowly giving way, and like the rotten boroughs of a past political period, the close preserves of the Colleges were being either extin- guished, or thrown open to public competition. But Oxford was still Conser- vative at heart. Leaders of the old school and their followers held the University pulpits, dominated Congregation, monopolized the best preferments, resisted to the best of their powers all local change, and were ready on provocation to ostracize unorthodox refonners for being, like Socrates, the corrupters of youth. Married Fellows were as yet unknown ; it had not yet become necessary to build whole suburbs of semi-detached villas to receive the feminine colonists of the future. But there was a stir and an agitation throughout the Academic world which the sense of changes, present and to come, had produced. University politics and polemics were, as always, of absorbing interest. Mansel and Goldwin Smith tilted against each other in debate before an admiring and competent academic audience. Oxford was, in fact, at war, — a war, iu is true, polite, polished, and courteous. The late Professor Jowett. BY LORD BOWEN. xlix Into this atmosphere, charged as it was with considerable personal electricity, Henry Smith was thenceforward absorbed ; for nearly thirty years, no more attractive, brilliant, or genial figure was to be found in the perturbed society of the University. Some happy combination of judgment and temper made him acceptable even to those with whose opinions he had nothing in common. He succeeded in being a politician, without wearing the obnoxious colours of a partisan. He had the great art of never pressing a victory home, and of bearing defeat with pleasant equanimity. His business powers, his modesty, his wisdom, and his entire freedom from egotism and dogmatic presumption, a delicate gaiety that never flagged, wit that sparkled without wounding, and which rose incessantly to real brilliancy, made him not merely an effective personage in the Oxford world, but universally acceptable in any society, whatever the shade of its opinions. His finished persiflage, his pleasant epigrams, wUl long be remembered, though the brightest conversation is often the most evanescent, and the finesse of wit, like a musical laugh, disappears with the occasion, and cannot be reproduced upon paper or in print. As by degrees his attainments were recognized, both in England and abroad, his influence at Oxford naturally deepened ; but neither within nor without the University did he grasp at opportunities for notoriety. Such power and authority as he possessed he held without an effort, without solicitation, apparently without any personal satisfaction in them. In ofiices of friendship he was constant ; in such public or civic duties as came in his way, assiduous ; no good or benevolent work ever needed a helping hand, but his was at its service, without ostentation, and without any expectation of personal advantage. He was a good speaker, without being a rhetorician ; his death, indeed, last week was hastened by a chUl caught or increased while he was addressing a gathering of agricultural labourers. A life like Henry Smith's, of exemplary moderation, far removed from even a suspicion of worldliness and vanity, is seldom found in these days in combination with intellectual powers and practical abUity on so considerable a scale. There are, no doubt, many nooks and corners in which at times may be seen flowering ' the wise indifference of the wise.' Students, divines, men of science or of letters, not seldom seem content to retu-e from the world, as if they had measured the time value of the things we most of us eagerly compete for, and were perfectly satisfied, of deliberate choice, to remain spectators of the fever of mankind. Some physical inaptitude, or some constitutional tendency, not unfrequently lies at the bottom of this apparently philosophic temper. Patient self-possession, and a sober estimate of the world and of g 1 RECOLLECTIONS OF PROFESSOR SMITH what it can give, are rarely found in a man who lives in constant contact with other men and their affairs, who shares in the interests of his generation, occupies himself with its business, and whose genius seems to bring high honour and success almost within his reach. Professor Henry Smith was not buried away from his fellow-creatures in literature, or study, or contemplation ; he was no recluse or invalid, but a man of the world, active, competent, social, ordy — not ambitious. Personal serenity of such a type is rather a classical than a modem virtue ; perhaps an age different to our own may yet regard it as one of the highest forms, not merely of intellectual, but of civic excellence. It is the characteristic of recent civilization, that in almost all its aspects it seems based upon a theory of personal competition. The prominent figures on every stage are the result of a struggle, not for existence, but for success. It is a contest which all seem satisfied to recognize as one of the conditions of ordinary life ; which constitutes the essence of our politics, of our commerce, of our political economy, of our laws of property themselves. In the general race to possess more than the average share of wealth, power, fame, it is, perhaps, a wholesome lesson to turn for a short breathing-time to the uneventful example of the life of a man of genius, who was fitted for most distinctions, if he had cared to seek them ; but who was unaffected by the universal fever, possessed his soul in perfect patience, and remained to the last content to discharge all the duties which Providence allotted to him, without affectation, and with that composure of soul to which great gifts are not always allied. The secret of the philosophic temperament, exhibited in this its most manly shape, is one which is not easy to explore ; but when the phenomenon is seen, its charm attracts us the more in proportion to its rarity. Essayists and moralists for the last two thousand years have preached it, and inculcated it ; some have gone so far as to boast of its acquisition, — its praise, certainly, is among all the prophets. Probably it is the product neither of Nature, nor of education singly, but of a happy, and of an admirable combination of the two. Among the many friends, acquaintances, admirers, whose thoughts have in the last few days been saddened or sobered by the unexpected death of a brilliant man of genius, there are none who will not readily accord to Professor Henry Smith the tribute of unaffected respect for what without extravagance may be termed his extraordinary powers of mind, his gentle and Laelian wisdom, and the sweetness of character which never made an enemy, lost a friend, or sought a personal advantage for itself But besides this and beyond this, it may not be out of place, before a personality in many ways so complete BY MB. STEACHAN-DAVIDSON. li fades into indistinctness, and a life ceases to be familiar to us which must hereafter be treasured rather in the memory of his contemporaries and friends than in the history of his time, to recognize in the Professor Oxford has lost that special type of wholesome and of manly virtue the growth of which is not much favoured by the rush and turmoil of these times. Great mental gifts can be found, when occasion demands them ; talents grow on every tree. But the serenity of heart which enables its possessor to wear the gifts of genius with sobriety, and to use them nobly and well, without seeking to expend them in the purchase of fame, or wealth, or of advancement, is a quality which modern society little cultivates, and seldom sees. III. RECOLLECTIONS BY ME. J. L. STEACHAN-DAVIDSON. The death of Henry Smith wUl be felt as the greatest loss which could have befallen Oxford. In him the University possessed a student whose know- ledge and genius were honoured throughout Europe. Of those amongst whom he lived few indeed could foUow him to the height of his scientific speculations. Most of us did not know enough to understand where and how he was working in the field of Mathematical Science. By us he is lamented as the wisest counsellor of the University, and as the delightful companion who gave life and charm to its society. Though his activity extended far beyond the limits of the University, he was very constant to Oxford. Since he took his degree he did not miss a single Term's residence. Ee-elected time after time to the Hebdomadal CouncU, his assistance was called for whenever any serious business required sound judgment or deUcate handling. His advice was generally followed, and if not, the neglect of it was almost always regretted in the sequel. In Henry Smith were united to a rare degree knowledge of business and knowledge of men. He seemed most thoroughly in his element when swaying and guiding his fellows. To every matter which he took in hand, he seemed to come with a fresh mind, throwing off all the multitude of concerns which beset him, and unburdened by care or anxiety. Then under the cover of his easy playful manner it would soon become manifest that he had grasped all the true points at issue, and was ready with a firm and wise decision. He always looked facts in the face, and strove hard to distinguish the diflScult from the impossible. To the more ardent spirits among his followers it was sometimes a matter of disappointment that he would not lead them to assaults which he saw to g2 lii RECOLLECTIONS OF PROFESSOR SMITH be fi-uitless. Though he could fight hard when the moment for fighting came, no one was more averse to multiply occasions of controversy. He saw things without passion and without prejudice, and laboured quietly and steadily for all that could advance the studies and promote the efiiciency of the University. In this spirit he accepted the thankless task of serving on the University Commission. It is the inevitable fate of such a body that their work is attacked at once by the criticisms of those who think that it has gone too far, and of those who would have had it go further. Henry Smith knew well that it was impossible to satisfy either the one party or the other. But it was a source of keen satisfaction to him to notice that when those who joined in complaining of the Commission came to propose alternative schemes they found that these divided them more than that to which they had objected. In the same way he was much gratified that it was only a minute point in the Commissioners' arrangements which was finally contested by the University. He claimed, and with justice, that when the Proctorial appointment of Examines was the only portion of the old constitution which was defended to the last, it was pretty clear that the more important changes were acknowledged to be wise and necessary. It may be interesting to note what Henry Smith thought of these greater changes. He fully appreciated the charms of the old system of celibate Fellow- ships, and never for a moment cherished the illusion that the new seven years' tenure could ever have the value and dignity of the old. But he felt that the old system could be practically worked only so long as the majority of Fellows were willing to take Orders and retire to a College Living in middle life. When it became evident that the University must either renounce the service of its most efficient members or be content to be served by laymen, he recognized that, at whatever sacrifice, a career must be provided into which a man could enter as his profession for life. Another important question often present to his mind was the effect of the College system on the life and teaching of Oxford. He felt the difficulties as keenly as many who urged radical changes ; but he felt likewise that it was worth making an effort to preserve this distinctive feature of the English Universities by transforming it to suit the new conditions. When in conver- sation he summed up the work of the Commission, it was, 'we have given a fresh lease of life to the College system.' He was not very sanguine that this system could be permanent, but he was convinced that it ought to have another chance, and that the best chance was secured to it by the reforms which he and his colleagues had effected. BY ME. STRACHAN-DAVIDSON. liii It is difficult to speak of the charm of his life and conversation. The light touch and happy play of mind with which he enlivened the most serious business, and softened the jarring of controversy, was a source of real power, and procured a ready acceptance for the wisdom of his practical suggestions. In social life the same qualities shone forth at every moment. It seems hardly credible to those who knew him best that a deep-seated disease had been sapping his life for years. His temper was always unruffled, his spirits always gay and easy, and his sympathy always ready. In the midst of a mass of business which would have absorbed any ordinary man he could always find time to attend to the interests and concerns of his friends. To cheer a sick friend with the sunshine of his presence, to be the protector of the children of those who were taken away, to lend a ready ear to the perplexed and a helping hand to those who had committed themselves by any foolish action — all such kindnesses seemed so easy and natural to him, that men claimed and accepted his benefits almost as a matter of course. He seemed to be good not in obedience to any external law nor as the result of any internal struggle, but because goodness was the simple outcome of his nature. His wit and gaiety were the delight of all who listened to him. It was not so much that he was a sayer of good things to be remembered and repeated — though of these too there was no lack — but the really charac- teristic feature of his talk was that its interest never flagged ; a certain flavour of freshness and originality pervaded it, and revealed itself even in his commonest remarks. To walk or ride with him was to enjoy a conversation in which not a sentence was commonplace. There was always some new light, some refine- ment or subtlety of thought or expression which gave a charm to the most ordinary topics. This eSect was due mainly to the keen and delicate temper of his mind, but partly also to the wonderful breadth of his knowledge and his interests. He knew the literature of Greece and Rome as if he had made their study the work of his life, whereas it was really the amusement of his leisure hours. He had the sincerest love for the classical writings and the most profound belief in their value. His retentive memory and delicate taste made his conver- sation on these topics a storehouse of interesting and instructive criticism. Though the resources of his own mind filled to overflowing every moment he could snatch for quiet study, yet he never shut himself up or held aloof from his fellows. There was absolutely nothing stem or forbidding about him. He seemed to take in the society of his friends the same pleasure which his presence imparted to them. In every relation of life there was in him the perfect ease liv RECOLLECTIONS OF PROFESSOR SMITH. and grace which flows naturally from complete and sufficing strength. The sweetness of his character and the perfect cordiality of his nature seemed to ofter all the rare gifts of his genius to minister to the happiness of his friends. His death leaves dark what was a ray of sunlight in the lives of many. He was entirely free from superstition. He held deliberately that the questions whose solution is hidden from man, and above all the prospect of death, should never be allowed to cast a shadow over the life and work of the present hoiu'. He believed that it became a man to live at his best and to labour at his best during every day allotted to him, even as though an endless succession of such days were in store. It was permitted to Henry Smith to give a bright example of his theory. Till within a week of his death he was teaching from his chair, attending to all the varied work of government and management for which he was responsible, and living a bright and happy life which shed cheerfulness and comfort on all around him. He always maintained, that there is no such thing as a necessary man, and that every place left vacant can be adequately filled. Of all that Henry Smith taught, this doctrine is the one which it seems most difficult to realize at this moment. [February, 1883.] NOTE BY MR. ALFRED ROBINSON. I HAVE been asked to give some account of the contested election in which Professor Smith was a candidate for one of the University seats in the House of Commons ; and I do this with much pleasure, because, although he was defeated, the amount and kind of support which he received in the contest show how much he was valued by men of all parties in Oxford, and how unique was his position in the University. In the spring of the year 1878 it became known that Mr. Gathorne Hardy, who was then one of the University representatives, was about to be summoned to the House of Lords. The rejection of Mr. Gladstone in 1865, and the defeat of Sir RoundeU Palmer by Sir John Mowbray in 1868, had proved that no one but a Conservative could win in a contest conducted upon the lines of political party. But it was thought by many persons that the Members fur the University ought to be chosen upon academical rather than upon political grounds, and ought to represent learning, science, and education, without special reference to party interests. THE ELECTORAL CONTEST OF 1878. Iv Professor Henry Smith was brought forward as a man most eminently quahfied to represent the University in this sense. The fact that he was a Liberal in politics of course was not disguised. He was indeed at this time not fully in sympathy with some of the Liberal leaders. The Eastern question then filled the political foreground, and Professor Smith, while disapproving of the general policy of the Conservative Government, thought that Lord Salisbury ought to be supported in maintaining against Russia in her dealings with Turkey, the rights of the neutral Powers and the general interests of Europe. Perhaps, also, Professor Smith was too critical, and too fond of looking at questions from every point of view to have ever made a first-rate party man. But still he belonged undeniably to the Liberal party, and he was not a man to be led by any waywardness, or by any love of fads or crotchets, into a position of political isolation. So his election by the University would no doubt have been a transfer of a seat from the Government to the Opposition side of the House, and this was the aspect in which the contest presented itself to the great majority of the voters. It was not, however, on a contest of this kind that Professor Smith's chief supporters wished to enter. In their view the special representation of the University in Parliament was useless if the University Members were to be party men of the ordinary type, without special qualifications for dealing with the questions with which the University was specially concerned, and theu' main object in bringing forward Professor Smith was that this view should be put before the constituency and the country With the arrangements for his own candidature Professor Smith had very little to do. An old custom of the University, which had been observed by Mr. Gladstone throughout his long tenure of his seat, precluded a candidate from issuing any address, or from making any speech to the electors. At an early stage in the proceedings Professor Smith was invited to stand, and he agreed to allow himself to be nominated, but he took no part in the initiation of his own candidature ; he was never present at the meetings of his committee ; and his supporters defrayed the expenses of the contest, declining a request which he made that he might at least be permitted to contribute to the subscription list. Professor Smith's Committee was formed in the month of April, 1878, and consisted of two sections. (1) The London Committee, the Chairman of which was Mr. Mountague Bernard ; and which had for its Vice-Chairmen the then Marquis of Tavistock, Mr. Goschen, Mr. Knatchbull Hugessen (afterwards Lord Brabourne), Mr. Dodson Ivi THE ELECTORAL CONTEST OF 1878. (now Lord Monk Bretton), Dean Stanley, the Dean of Canterbury, the Dean of Durham, and the late Sir Benjamin Brodie ; and for its Secretaries, Mr. BuUer of All Souls, Mr. Ilbert, late Bui-sar and Fellow of Balliol, Mr. Pope, formerly Fellow of Lincoln, Mr. Robertson, Fellow of Corpus, Mr. A. L. Smith, then Fellow of Trinity. (2) The Oxford Committee, having for its Chairman the Dean of Christ Church (Dr. Liddell) ; for its Vice-Chairmen, the then President of Corpus and Archdeacon Palmer ; and for its Secretaries, Mr. Crowder, Bursar of C. C. C, Professor Green, Mr. Jackson, Fellow and now Rector of Exeter, Mr. Monro, Fellow and now Provost of Oriel, Mr. Papillon, Fellow of New College, Mr. Salwey, Student of Christ Church. It is not likely that any of these persons were under the illusion that their cause was going to win. The Conservative feeling of the constituency was soon found to be so strong that under no circumstances could any one but a Conser- vative have been elected. And at this time the excitement of the two political parties with reference to the Eastern Question much increased the difficulties with which Professor Smith's Committee had to contend. On the one hand, a large section of the constituency saw in him only an opponent of the Govern- ment which was patriotically defending British interests against Russia. And on the other, some well-known Liberals considered that he was too lukewarm in his censure of the Party and of the Ministry which they associated with the notorious atrocities in Bulgaria. Opposition of the former kind, which insisted that the representative of the University of Oxford must be a supporter of a Conservative government, it was impossible to disarm. But an effort was made to conciliate the critics and opponents who belonged to the Liberal ranks. Professor Smith was requested by his Committee to put forward some definite statement of his views on the Eastern Question, and in the following letter to a member of his Oxford Committee he complied with this request. Dear , ^^ '^' ^'''- I am well aware that the custom of the University imposes a great measure of reserve upon any candidate for the honour of representing it in Parliament. But I do not think that I shall be departing from a tradition, which I am most anxious to see maintained, if I venture to write a few lines to you in explanation of the views which I entertain with regard to the Eastern Question. There has been much in the foreign policy of the Government during the last two years of which I cannot approve. I think that they should have recognised, at a far earlier period than they did, that the condition of the Christian Provinces of Turkey NOTE BY MR. ALFRED ROBINSON. Ivu had become unendurable, and that the maintenance of the status qiuo was no longer possible. A grave, but long foreseen, emergency had arisen ; and this country should have been prepared with a well-considered policy to meet it. Instead of this, the Ministry seem to me to have drifted with the stream of events, until at last they find themselves in a position in which it is immeasurably more difficult, than it would have been twelve months ago, to assert the right of the neutral powers to have a decisive voice ■ in the settlement of a question affecting such vast European interests. Looking at the most recent events, I have to express a general concurrence with the main tenor of Lord Salisbury's Despatch ; and I have observed, with great satisfaction, that it has been received with cordial approval by the Liberal Press on the Continent. Interpreting that document, as I think I am justified in doing, by the light thrown on it by Lord Salisbury's own conduct at the Conference of Constantinople, I do not perceive in it any intention to restrict the liberties to be granted to the Christian subjects of the Porte ; but I regard it as a protest in favour of the recognition of inter- national obligations, and against any attempt on the part of Russia, to dispose of the Eastern Question in her own way. For the reasons which I have stated, I feel that I could not enter Parliament, except upon the condition of preserving the right to form while there an independent judgment with regard to the future action of the Government in these important matters. If there is a war party in England, I have no sympathy with it : but I am not for peace at any price ; and, if any of the great interests of the country should be endangered, I should hope to see aU Englishmen, without distinction of party, united in defending them. Believe me to remain. Very faithfully yours, Henry J. S. Smith. P.S. — You are at liberty to make any use which you may think fit of this reply to your letter. This letter removed some of the misapprehensions as to Professor Smith's position, and probably produced some effect upon the canvass. Promises of support were received from some of the Liberals who had originally stood aloof, including one from Mr. Gladstone, which arrived a few days before the opening of the Poll. His example, however, was not imitated by all his followers, a few of whom, more Gladstonian than their chief, remained neutral to the last. May 13th was fixed for the nomination. On that day the two candidates were proposed to the House of Convocation in Latin speeches — Professor Henry Smith by the Dean of Ch. Ch., Mr. J. G. Talbot by the President of St. John's. The Dean dwelt upon the scientific and literary qualifications of Professor Smith, his abUity in business and in debate, and his suavity and fairness of judgment, which conciliated the regard of all. He recommended him to the electors as a man whom the Ministry of the day had entrusted with the most weighty h Iviii THE ELECTOEA.L CONTEST OF 1878. affairs, and as one ' unice idoneum qui ipse academicus academicos suos in Parlia- mento reprsesentet.' The President of St. John's, in nominating Mr. Talbot, made some kindly remarks on the undesirableness of withdrawing Professor Smith from the Professorial duties and from the sciences which he adorned. Immediately after the nomination the Poll opened, and under the Act governing University elections it was not to be closed until the 17th, unless either of the candidates were withdrawn in the meantime. The electors could vote either in person or by voting papers. At the time when the voting began Professor Smith's Committee had received less than a thousand promises from a constituency numbering more than four thousand members, and the last hope of the most sanguine of his supporters had dis- appeared. It was, however, thought best that all the votes should be recorded, in order that the amount and kind of support with which his candidature had been received might be accurately measured and generally known ; so the polling was continued daily for five days in all. At the close Mr. J. G. Talbot was declared to be elected, the numbers being — Talbot, 2687 ; Smith, 989. Defeated by a majority of more than two to one. Professor Smith's Committee might to some extent console themselves with the thought that they had conducted the contest with great economy. The expenses amounted to about £420, the chief item being the bills for advertising the lists of supporters in the chief London papers. This sum was less than half of what was believed to have been spent on the Liberal side in each of the two preceding Oxford elections. Still more consolatory was the analysis of the Poll Book, which was pub- lished soon after the result of the election was declared. This proved that the majority of the electors and the working staff of the University had been ranged under opposite banners. The following table shows how certain sections of the constituency had voted : — SMrrH. Talbot. Abstained &om Toting. Heads of Colleges, including two acting Heads Professors, Readers, and University Teachers 10 28 91 159 152 lO II 39 82 ii-j 3 6 13 53 44 Tutors and Lecturers of C!olleges and Halls Fellows of Colleges, resident and non-resident *Residents Besidents — Members of Congregation qualified by residence — i. e. all electors who were in residence Oct. 1876 — Oct. 1877, including the parochial clergy in Oxford and others not engaged in University work. I NOTE BY MR. ALFRED ROBINSON, Kx Whether this table points to any practical conclusion or not may be doubted. That the Members for the University should be chosen by those who are identified with it as the place of their work or residence in the present, and not merely of their education in the past, may seem reasonable, but no change could possibly be made which would reduce the constituency to less than one-tenth of its former number ; and perhaps the special representation of the Universities in Parliament is more likely in the future to be abolished than to be reformed. But whatever inferences of this kind might be drawn from the result of the election, the Poll Book was unequivocal in its recognition of Professor Smith's personal qualifications and eminence. And even the numbers set forth in the foregoing table, emphatic as they are, do not fully express the estimation in which he was held by that portion of the constituency in the midst of which he had lived, and with which he had been officially connected. For among those who abstained from voting there were some who remained neutral, in spite of their high appreciation of his claims, because they thought that a seat in Parlia- ment would be incompatible with his Oxford work ; and there were others who on ordinary occasions would have been ranked among his warmest supporters, but were unable at this time, when the foreign policy of the country filled the political horizon, to vote for a man who was variously criticised as going too far, or not far enough, in support of, or in opposition to, the Government of the day. But, in spite of these abstentions, the preponderance of opinion in the working staff of the University was clearly marked, and was most significant. It may be confidently said that no other man could have enlisted at this time among his supporters in a Parliamentary contest so many of the men who were identified by their position or occupation with Oxford ; and it may be doubted whether in any of the controversies, political and academic, which have divided the University at various times in its history, so many of its resident graduates have ever enrolled themselves upon one side. h 2 INTRODUCTION TO THE COLLECTED MATHEMATICAL PAPERS OF HENRY J. S. SMITH. THE present volumes contain all the mathematical papers published by the late Professor H. J. S. Smith in his lifetime, as well as those which were in the press or which had been written out for printing. The reader is therefore in possession of all that he had already published, or had wholly or partially prepared for publication at the time of his death. The arrangement of the papers is strictly chronological, the order being that of the date of reading or publication. The only partial exception is the Report on the Theory of Numbers, which is printed as a whole, although several other papers which follow it were published in the six years during which it was in progress. It is possible therefore by merely glancing over the titles of the papers to trace the course of Professor Smith's mathematical studies and tastes. The first two papers, written in 1851 and 1852, show that his mind was then occupied by Geometry. Within three years he published his first paper on the Theory of Numbers, consisting of a characteristic proof of Fermat's theorem that every prime number of the form 4n + 1 is the sum of two squares. From this time until 1867 the printed papers relate almost exclusively to the Theory of Numbers. Then follow a number of geometrical papers. In the last years of his life he was occupied pi'incipally with the subject of Elliptic Functions. It will be seen therefore that his work falls into three distinct groups : (i) Geometry, (ii) Theory of Numbers, (iii) Elliptic Functions. From the fact that the two earliest papers relate to Geometry we may infer that this was the sub- ject which originally proved most attractive ta him. The first of these papers was written in the year in which he obtained the Senior Mathematical Scholarship, and only a little more than a year after his election to a Fellowship at Balliol. It would seem that about 1853 he commenced the study of the Higher Arithmetic, Ixii INTRODUCTION. a subject which engaged his almost undivided attention for many years, and which was never afterwards quite absent from his thoughts. The short notes which bear the dates of 1854 and 1857 show the tendency of his mind at this time : and in 1859 the first part of his Report on the Theory of Numbers was contributed to the British Association. The subsequent instalments appeared in the annual volumes of the Association for 1860, 1861, 1862, 1863, and 1865. These reports, which contain in a very condensed form the result of an immense amount of research, are models of clear exposition and systematic arrangement. Besides the accounts there given of the work of others, many of the paragraphs contain results of his own. These original contributions are not, however, noted as such, and they can only be detected by those who are already well acquainted with the details of the subjects to which they belong. During the preparation of this Report he carried out elaborate researches of his own in several important branches of the Higher Arithmetic. The principal investigations undertaken at this time, which were completed for publication, relate to systems of indeterminate linear equations and congruences and to the orders and genera of ternary quadratic forms containing more than three indeter- minates. These memoirs appeared in thp Philosophical Transactions for 1861 and 1867. He also contributed several shorter papers to the Proceedings of the same Society, which indicate much more extended investigation in the same field : one especially (No. xviii, vol. i.) consists merely of a brief statement of results which were obtained by means of a very long and delicate analysis. A considerable part of the last instalment of the Report is concerned with arithmetical formulae derived from Elliptic Functions, and it seems likely that it was in this way that he was first attracted to this Theory ; for his first published paper on the subject (No. xvi, vol. i.) bears the date 1866. The x'emaining papers in- cluded in the first volume relate to Geometry, principally homogi'aphic figures. In the second volume (1869-1883) there is more Elliptic Functions and less Theory of Numbers : but the sequence of the papers no longer afibrds an indication of the author's train of thought : for, in the later years of his life, he was frequently compelled, by various circumstances, to leave the subject upon which he was engaged, in order to prepare for publication theorems and demon- strations that formed part of the many unfinished investigations stored up in his note-books. The first paper in the second volume was a prize memoir for which, con- jointly with another memoir, the Steiner prize of the Berlin Academy was awarded. The subject was announced in 1866, and the memoirs were to be sent I INTRODUCTION. Ixiii in, each designated by a motto, before March 1, 1868*. Four were received, and the prize of six hundred thalers was divided between Professor Smith and Dr. Hermann Kortum, of Bonn, the two memoirs being regarded as of equal merit. The report on the memoirs received, which was laid before the Academy by Professor Kummer on July 2, 1868, contained the following remarks relating to that sent in by Professor Smith : ' Die vierte, in franzcisisclier Sprache abgefasste Preisschrift mit dem Motto : " Haud facilem esse viam voluit " fiihrt den Titel : " Memoire sur quelques problfemes cubiques et biquadratiques," und ist in drei Abschnitte eingetheilt. Der erste Abschnitt beschiiftigt sich mit der Theorie des Imaginaren in der Geometrie, der zweite enthalt verschiedene Methoden, die gemeinsamen Puiikte zweier durch ihre Elemente gegebener Kegelschnitte mittels des Lineals, des Cirkels und eines festen Kegelschnitts zu censtruiren, in dera dritten Abschnitte endlich lost der Verfasser ausser einigen andern sogenannten kubischen und biquadratischen geometrischen Aufgaben namentlich das speciell in der Preisfrage hervorgehobene, die Curven vierten Grades betreffende Problem. Die ganze Arbeit zeichnet sich durch iibersichtliche und gystematische Behandlung des Stoffes aus. Der Verfasser macht bei seinen Constructionen, wie es in der Preisfrage verlangt wird, nur von den einfachsten erforderlichen und ausreichenden Hilfemitteln Gebrauch, aber bei den Constructions-Methoden selbst hat er mehr auf gedankliche als auf praktische Einfachheit, mehr auf die vollstandige Darlegung aUer Gesichtspunkte als auf die Ausfiihrung aller einzelnen Operationen sein Bestreben gerichtet. Dadurch ist es ihm gelungen, im zweiten Abschnitte das an sich diirftige und trockene Material in gediegener und interessanter Weise zu verarbeiten und im dritten Abschnitte die specielle dort behandelte Frage mit allgemeineren zu verknilpfen. Fast uberall lasst die Arbeit zum Vortheil fiir ihren wissenschaftlichen "Werth deutlich erkennen, dass der Verfasser zu seinen umfassenderen Untersuchungen durch alge- braische Betrachtungen gelangt ist, deren genauer Zusammenhang mit dem Gegenstande der Preisfrage schon in deren Formulinmg enthalten ist t.' The origin of the long memoir on the Theta and Omega Functions — the last paper but one in the second volume — was as follows. At the end of 1873, or the * The announcement of the subject was made in the following terms : ' Fiir diejenigen geometrischen Probleme, deren algebraische Losung von Gleichungen von hoherem als dem zweiten Grade abhangt, fehlt es noch an der Feststellung der zur constructiven Losung derselben erforderlichen und ausreichenden fundamentalen Hilfsmittel, so wie an den Methoden zur systematischen Benutzung dieser Hilfsmittel. ' Indem die Akademie die Frage, die sie stellt, auf die Probleme beschrankt, welche auf kubische Gleichungen fiihren, wiinscht sie, dass wenigstens an einer Anzahl von speciellen Beispielen gezeigt werde, wie diese Liicke in dem Gebiete der constructiven Geometrie ausgefiillt werden kbnne. Namentlich verlangt sie die vollstandige Losung des folgenden Problems : ' Wenn dreizehn Punkte in der Ebene gegeben sind, so soUen durch geometrische Construction diejenigen drei Punkte bestimmt werden, welche mit den gegebenen zusammen ein Sy-stem von sechzehn Durchschnittspunkten zweier Curven vierten Grades bilden. ' Bei der Losung sind die Falle zu beriicksichtigen, . in welchel) einige der dreizehn Punkte imaginar und demgemass nicht als individuelle Punkte, sondem als Durchschnittspunkte vorgelegter Curven gegeben sind. Gewiinscht wird ferner, dass sammtliche geometrische Constructionen durch die entsprechenden algebraischen Operationen erlautert werden.' t Mmiatgberichte for 1868, p. 420. Ixiv INTRODUCTION. beginning of 1874, when I was passing through the press the Tables of the Theta Functions which I had calculated in connexion with a Committee of the British Association, I asked Professor Smith, who was a member of the Committee, if he would contribute an Introduction to the volume. He replied that he did not see his way to writing anj^thing appropriate to the tables themselves, but that he 'could say something with respect to the constants at the head of the pages.' These constants were K, K\ E, J, J', &c., the numerical values of which were given for every minute of the modular angle. The memoir grew in extent, and was subject to frequent interruptions ; in fact a number of other papers were written and published during its progress. These papers were generally called into existence by special cir- cumstances unconnected with the memoir, but a few of them, and especially the Notes on the Transformation of Elhptic Functions (Nos. xli, xlii, vol. ii.), which immediately precede it in the volume, arose directly out of it. The first two of these Notes were given to me in the summer and autumn of 1882 for the Messenger of Mathematics, and appeared in the numbers from August to November of that year. The remaining Notes were printed after his death from a draft manuscript which he had shown to me, and explained in some detail, in October, 1882. The memoir itself, with the Notes that were con- nected with it, formed the principal new work upon which he was engaged from the time of its commencement untU his death : most of the other papers, published in the interval, containing results which were mainly derived from his earlier investigations. It was left incomplete : Arts. 1-31 (pp. 415-484) had been passed for press : Arts. 32-48 (pp. 485-535) had been revised, and Arts. 49-73 (pp. 535-585) were in type in quarto pages and had been partially corrected. The succeeding Articles up to Art. 88 inclusive were in type in octavo slip, and had been pai-tially corrected in this form *. The last two Articles (89 and 90) are printed from a manuscript found among his papers, and which he had marked as following on after Art. 88. I believe that no more was written, even in draft. The figures which occur in the Memoir had not been drawn. The object of Professor Smith's first paper on Elliptic Functions (No. xvi, * The whole of the Memoir was originally set up all in octavo slip, and remiined in this form for a long time, during which it was greatly altered and extended. It was reset in quarto pages during 1881 and 1882, and passed for press in this form. It had been intended that it should appear as an Introduction, but it was finally decided that it should follow the tables with the title ' Memoir on the Theta and Omega Functions.' INTRODUCTION. Ixv vol. i.) was, as stated in the first paragraph, to enunciate and demonstrate a fundamental theorem, the nature of which had been indicated in a letter, written in 1845, from Jacobi to M. Hermite, in which he mentioned that he used it as the starting-point in his Konigsberg lectures. Jacobi died in 1851, and as the theorem referred to had never been published. Professor Smith reproduced it, in 1866, in this paper. Guided by Jacobi's suggestion, he multiplied together four general Theta series and expressed the product as the sum of four terms, each of which was the product of four Theta series with different arguments. From this theorem he derived, as indicated by Jacobi, all the principal results of Elliptic Functions, either as particular cases or as simple corollaries. In 1881 the first volume of the Collected Works of Jacobi was issued, and his Konigs- berg lectures on Elliptic Functions were there printed for the fii'st time. By com- paring them with Professor Smith's paper it will be seen that, although the theorem itself is of course essentially the same, still there are differences in the mode in which it is presented which enhance the interest of the latter. Professor Smith treated the question with great generality, and with absolute precision, and this short paper is very characteristic of his style of work. At the meeting of the London Mathematical Society on January 8, 1879, Professor Cayley communicated to the Society the theorem A:^^'2snasn/3sn7sn(J — ^^cnacn/Scn^cn^ -1- dnadnjSdnydncJ — F^ = 0, where a, /3, y, S are any quantities whose sum is zero. Professor Smith, who was present at the meeting, remarked that it was a special case of a theorem relating to the multiplication of four Theta functions, and at the next meeting in February he communicated to the Society the general Theta Function formulae which dominate all results of this class. This paper {No. xxxviii, vol. ii.), which is supplementary to that of 1866, was written out from notes which he had had by him since that date. The paper on the conditions of perpendicularity in a parallelepipedal system (No. xxxii, vol. ii.) was written in response to a request from his friend Professor Maskelyne, who was seeking for a general treatment of a problem which, in the particular case of its application to crystallography and the dis- tribution of molecules in a crystal, was of paramount importance. The circumstances connected with the publication of the memoir which concludes the second volume require a more extended notice. In February, 1882, he was surprised to see in the Comptes Rendus that the subject proposed by the French Academy for the Grand Prix des Sciences Math^matiques was the decom- Ixvi INTRODUCTION. position of a number into five squares *. His feelings in the matter are shown by the following extracts from letters to myself In the first, dated Oxford, February 17, 1882, he wrote — 'The Paris Academy have set for their Grand Prix for this year the theory of the decomposition of numbers into five squares, referring to a note of Eisenstein, Crelle, vol, xxxv, in which he gives without demonstration the formulae for the case in which the number to be decomposed has no square divisor. In the Royal Society's Proceedings, vol. xvi, pp. 207, 208, I have given the complete theorems, not only for five, but also for seven squares : and though I have not given my demonstrations, I have (in the paper beginning at p. 197) described the general theory from which these theorems are corollaries with some fulness of detail. Ought I to do anything in the matter ? My first impression is that I ought to write to Hermite, and call his attention to it. A line or two of advice would really oblige me, as I am somewhat troubled and a little annoyed ;' and in the second, of date February 22, he proceeded, 'You see I take your advice entirely upon the point that he ought to be written to. The worst of it is that it would take me a year, and a hundred pages, to work out the demonstrations of the paper in the Royal Society's Proceedings.' The following reply was received from M. Hermite : MON CH£B MONSIEITB, Aucun des membres de la commission qui a propose pour sujet du prix des sciences math6matiques en 1 882 la demonstration des th^orfemes d'Eisenstein sur la decomposition des nombres en cinq carrfe n'avait connaissance de tos travaux contenant depuis bien des ann^es cette demonstration et dont j'ai pour la premifcre fois connaissance par votre lettre. L'embarras n'est point pour vous, mais pour le rapporteur des memoires envoy6s au concours, et si j'etais ce rapporteur je n'hesiterais pas un moment k faire d'abord I'aveu complet de I'ignorance ou il s'est trouve de vos publications, et ensuite k proclamer hautement que vous aviez donn^ la solution de la question propo86e. Une circonstance pourrait 6ter tout embarras et rendre sa t4che facile autant qu'agr(fable. S'il avait en efifet k rendre compte d'un memoire adresse par vous-mSme dans lequel vons rappelleriez vos anciennes recherches en les com- pietant, vous voyez que justice vous serait rendue en m6me temps que les intentions de I'Acadi'mie * The subject of the prize for 1882 had also been announced a year previously, but the notice had then escaped his attention. The following are the terms of the announcement : Grand Prix des Sciences Mathematiques. (Prix du Budget.) Question proposee pour I'annee 1882. L Acaddmie propose pour sujet du prix la TMorie de la decomposition des nombres entiers en une somme de cinq carrea, en appelant particulifcrenjent I'attention des concurrents sur les r^sultats extrfemement remarquables enonc^s sans demonstration par Eisenstein dans une Note ecrite en langue fran9ai8e au Tome 35 du Journal de Mathematigues de Crelle (p. 368, annee 1847). ' Le prix consistera en une medaille de la valeur de trois mille francs. 'Les Memoires devront 6tre remis au Secretariat avant le ler juin 1882; ils porteront une epigraphe ou d6vise repetee dans un billet cachete qui contiendra le nom et I'adresse de I'auteur. Ce pli ne sera ouvert que si la pifece k laquelle il appartient est couronnee.' (Comptes Rendiis, vol. xcii. p. 622, March 14, 1881, and vol. xciv. p. 330, Feb. 6, 1882.) INTRODUCTION. Ixvu seraient remplies puisqu'on lui annoncerait la solution complfete de la question propos^e. Jusqu'ici je n'ai pas eu connaissance qu'aucune pifece ait et6 envoyee, ce qui s'explique par la direction du courant mathematique qui ne se porte plus maintenant vers I'arithm^tique. Vous Stes seul en Angleterre k marcher dans la voie ouverte par Eisenstein. M. Kronecker est seul en Allemagne ; at chez nous M. Poincar6, qui a jet6 en avant quelques id^es heureuses sur ce qu'il appelle les invariants arithmetiques, semble maintenant ne plus songer qu'aux fonctions Fuchsiennes et aux equations differentielles. Vous jugerez s'il vous convient de r^pondre k I'appel de I'Academie k ceux qui aiment I'Arithmetique ; en tout cas soyez assur6 que la commission aura par moi connaissance de vos travanx si elle a se prononcer et k faire un rapport k I'Academie sur des m^moires soumis k son examen . . . Je vous renouvelle, mon cher Monsieur, I'expression de ma plus haute estime et de mes sentiments bien sincferement d6vou6s. CH. HERMITE. Pabis, 26 F^vrier, 1882. In consequence of an accident when riding, Professor Smith had been confined to his sofa for some weeks ; but, as far as his strength permitted, he had been working steadily at subjects connected with the memoir on the Theta and Omega subjects, which he was very reluctant to lay aside. Nevertheless, he thought it his duty to accede to the suggestion of M. Her- mite, and bring his demonstrations before the Academy in the form of a memoir sent in for the concours. For a whUe he divided his spare time between Elliptic Functions and the work connected with the prize subject, but in April he wrote : ' I fear I cannot let you have the Transformation papers before the end of June. As I foresaw, getting the quadratic forms of n indetermlnates into my mind again, putting my proofs into a rigorous form, and writing them out, will take up every moment till the end of May (the paper has to be In Paris by June 1). My sole reason for taking this trouble is that sooner or later I should have had to do it unless I was to allow my demon- strations to perish.' Professor Smith died on February 9, 1883, and it was not till nearly two months after his death (at the meeting of the Academy on April 2) that the report of the Commission was announced, two prizes being awarded, one to Professor Smith and one to M. Minkowski, of Kdnigsberg. The following Is the text of the report : Grand Prix des Sciences Math6matiques (Prix du Budget). (Commissaires : MM. Hermite, Bonnet, Bertrand, Bouquet ; Jordan, rapporteur.) L'Acadtmie avait propose pour sujet de prix la ' Thcorie de la d^compos-ition des nombres entiers en une somme de cinq carr68,' en appelant particulierement I'attention des concurrents sur les r^sultats extrfemement remarquables ^nonces pans demonstration par Eisenstein dans une Note dcrite en langue fran9ai8e an tome 35 du Jtywmal de Mathematiques de Crelle, p. 868, annce 1847. i 2 Ixviii INTRODUCTION. Ce problfcme semble assez restreint au premier abord ; mais on avait lieu de penser que les thterimcs obtenus par cet illustre g^ombtre sYtaient offerts i lui comma consequences derniferes d'une loDgue sorie de recherclies, oil devaient se trouver combintes les notions d'ordre et de genre, dtablies par Gauss pour les formes binaires, et transportees par Eisenstein dans le domaine des formes temaires, celle de la dentili, qu'il avait iutroduite pour la premiere fois, enfin les m6thodes iniinit^simales de Dirichlct. L'Acadimie 6tait done fondte k espirer que ce voyage de d(jcouvertes impos6 aux concur- rents 4 trovers une des regions les plus int^ressantes et les moins explor^es de TArithm^tique produirait des r^ultats f^couds pour la Science. Cette attente n'a pas iU: tromp^. Trois M^moires ont 6te transmis au Concours ; ils portent les ^pigraphes suivantes : No. 1. Quotque quibusque modis possint in quinque resolvi Quadratos numeri, pagina nostra docet. No. 2. Felix qui potuit rerum cognoscere causas 1 No. 3. Rien n'est beau que le vrai ; le vrai seul est aimable. Le M^moire No. 2 montre chez son auteur des connaissances t'tendues et renferme plusieurs r^sultats int^resEants ; mais la question pos4e par TAcad^mie ne s'y trouve m6me pas abordee. La Commission a done principalement concentr6 son etude sur les deux autres M^moires. Tous deux sont des oeuvres considerables, oil se trouvent exposes d'une manifere magistrale plusieurs des points fonda- mentaux de la th^orie des formes quadratiques. Les formulas relatives a la decomposition en cinq carr^s n'y figurent que comme consequences trfes particuliferes des principes gent'raux. II est d'ailleurs aise de discerner dans ces deux Memoires, k travers les differences d'exposition, une singuli6re identite dans la filiation des idees, au point qu'il serait difficile de signaler dans I'un d'eux une notion ou un theorfeme important qu'on ne retrouvat pas dans I'autre, et que, pour eviter les redites et faire mieux ressortir les nuances qui les separent, nous devions les analyser simultanement. L'auteur du Memoire No. 1 montre tout d'abord qu'4 une forme quadratique quelcouque on pent associer une serie de formes adjointes * ; la valeur numerique du plus grand commun diviseur des coefficients de ces diverses formes et leur ordre de parity servent de base k une distribution en ordres de mfeme determinant. L'auteur du Memoire No. 3 ne parle pas de ces formes adjointes, si ce n'est de la premifere, que Gauss avait deji definie ; mais il considfere la serie de leurs coefficients, ce qui lui donne un resultat identique au prectdent. La marche suivie dans les deux Memoires est d'ailleurs la nieme et consiste k transformer la forme proposee en une autre equivalente, telle que son residu par rapport k un module donne soit ramen^ k une expression canonique. Cette expression canonique contient encore des coefficients indetermines dont la valeur dependra de la manifere de conduire les calcnls ; mais de quelque fa^on que Ton opfere, en partant d'une forme donnee, certaines combiuaisons de ces coefficients conserveront toujours un caractfere quadratique determine par rapport aux nombres premiers qui divisent le determinant et par rapport aux nombres 4 et 8. L'ensemble de ces caractferes, invariables pour toutes les formes d'une mSme classe, definira le genre. Ainsi que Gauss I'avait dej4 signaie pour les formes binaires, en insistant tout particuliferement sur ces circonstances, qui sont pour I'Arithmetique du plus haut interet, toutes les combiuaisons de caracterea ne sont pas admissibles. Les deux auteurs indiquent d'une fafon precise les conditions que doit remplir une semblable combinaison pour correspondre k un genre reellement existant. lis passent ensuite k la recherche du nombre des solutions des congruences du second degr6 k plusieurs inconnues. Cette question se lie intimement aux pr6cedentes. La methode elegante fondee sur I'emploi de la resolvante de Lagrange, par laquelle elle est traitee dans le Memoire No. 3, m6rite * Ces formes avaient deji et6 considerees par M. Darboux dans le Jvwmal de Liouville. INTEODUCTION. Ixix une mention particulifere. L'Auteur toonce ensuite cette proposition, dont il est facile de r^tablir la demonstration : Benjux classes de formes qui appartiennent au meme genre sont congrues par ra])2>ort & un module quelconque. Cette nouvelle definition du genre, deja fonnulee d'ailleurs par M. Poincar6, a I'avantage de s'etendre imm^diatement aux formes d'ordre superieur au second. Les deux auteurs s'occupent ensuite de la representation des nombres par une forme quadratique k n variables. lis montrent, en g^n^ralisant une methode de Gauss, que cette recherche revient k celle de la representation d'une forme quadratique a n — 1 variables. Abordant ensuite ce dernier problfeme, ils font voir comment I'ordre et le genre de la forme representee peuvent se deduire de I'ordre et du genre de la forme qui la represente. Les resultats precedents leur permettent de ramener la recherche de la densite des representations d'un nombre donne par I'ensemble des formes d'un meme genre k celle de la densite d'un genre donne. L'application des methodes de Dirichlet a foumi la solution de ce probieme k I'auteur du Memoire No. 1 pour les formes quatemaires ; k celui du Memoire No. 3 pour les formes k un nombre quelconque de variables dont toutes les adjointes sont des formes impaires. Mais chacun d'eux, presse par le temps, n'a donne la demonstration de ses resultats qu'autant qu'il etait necessaire pour resoudre le problime pose par 1' Academe. Tous les deux le ramfenent k la sommation d'une eerie infinie, ^^ ^m,'' mr fort analogue k celle que Dirichlet avait rencontree dans son ceifebre Memoire sur les applications du Calcul infinitesimal k la Theorie des nombres. L'auteur de Memoire No. 3 s'arrete k ce point ; celui du Memoire No. 1 donne sans demonstration le resultat de la sommatiou, d'oii decoulent immediatement les theoremes d'Eisenstein. De m§me que nous n'avons pu separer ces deux beaux Memoires dans la courte analyse qui precede, nous ne saurions les presenter I'un sans I'autre aux suffrages de I'Academie. Tous deux en sont egalement dignes. lis font faire un pas considerable k I'Arithmetique, en fixant d'une maniere definitive la theorie de I'ordre et du genre dans les formes quadratiques. Le talent deploye par les auteurs nous est d'ailleurs garant qu'ils sauront mener k terme les questions difficiles qu'ils ont du traiter un peu hativement ci la fin de leur travail. Dans I'impossibilite ou elle se trouve de mettre I'un d'eux au second rang, la Commission k I'una- nimite emet le vceu que I'Academie accorde k chacun d'eux la totalite du prix, si elle le juge possible. Nous devons faire observer, en terminant, que le Memoire No. 3 est ecrit en allemand, contrairement k I'une des conditions du programme. L'auteur s'en excuse dans sa Preface, en disant que le temps lui a manque pour faire la traduction de son Memoire. Nous n'avons pas pense qu'il y eut lieu de repousser a priori, pour une irregularite de forme, un travail de cette importance. Mais, tout en I'accueillant, k titre exceptionnel, I'Academie devra faire toutes reserves pour l'application des regies ordinaires aux concours k venir. L'Academie adopte les propositions de la Commission et decide qu'elle decernera deux prix de m6me valeur aux auteurs des Memoires inscrits sous les Nos. 1 et 3. Conformement au Rfeglement, M. le President procfede k I'ouverture des plis cachetes qui accom- pagnent ces Memoires et proclame pour le No, 1 le nom de M. J. S. Smith, professeur k I'Universite d'Oxford, et pour le No. 3 nom de M. Hermann Minkowski, etudiant de Mathematiques k I'Universite de Konigsberg. lb will be seen that in this report, which has been reproduced in its entirety, no mention is made of Professor Smith's previous pubUcations, nor is there even a reference to his having completed Eisenstein's formulae for five squares, and \xx INTRODUCTION. given the corresponding formulae for seven squares, more than fifteen years before : in fact, the report shows that the writer regarded Professor Smith's memoir as perfectly new work called into existence by the prize competition. Under these circumstances Miss Smith, as the representative of her brother, wrote to M. Her- mite recalling his attention to the expression in his letter of February 26, 1882, ' En tout cas soyez assur^ que la commission aura par moi connaissance de vos travaux si elle a se prononcer et h, faire un rapport k I'Acaddmie sur des m^moires soumis k son examen,' and expressing the hope that he would give the explanation that had become necessary. M. Hermite replied that the omission of which she complained was an error which was due to absolutely involuntary forgetfulness (' ce tort ne consiste que dans un oubli, qui a 6t6 absolument invo- lontaire') ; but he made no further statement of any kind. The award of the prize gave rise however to a good deal of comment in the Paris newspapers. The Academy was blamed for having been unaware of work published by the Royal Society in 1868, and it was pointed out that the award was necessarily unsatis- factory, in spite of Professor Smith himself having sent in a memoir, as any other competitor might have availed himself of the indications contained in his published writings. The striking identity between the first and third memoirs, which is emphasized in the report, gave rise to the statement, which appeared in the newspapers, that this had actually taken place. In consequence of these criticisms M. Bertrand made certain explanations at the meeting of the Academy on April 16, 1883. The proceedings commenced with the reading of an appreciative obituary notice of Professor Smith by M. Camille Jordan, in which special reference was made to his arithmetical researches. The account then proceeds : M. Bertrand demande k TAcad^inie la permission d'ajouter quelques mots k la lecture qu'elle vient d'entendre. ' La Commission charges de proposer le snjet du prix de Math^matiques avait demands aux concurrents I'dtude d'un thiorfeme 6nonce, il y a prfes de quarante ans, par I'illustre gtomfetre Eisenstein, enlev6 k la science avant d'en avoir public la demonstration. ' Un seul Memoire depuis la mort d'Eisenstein avait et6 consacrd k cette difficile question : il 6tait de M. Smith et, comme celui d'Eisenstein, coutenait I'^nonce seulement des resultats priiicipaux. Si le coucours propose par I'Acad^mie n'etait pas venu reporter I'attention de M. Smith vers ces lecherches d^ji anciennes, il n'aurait, de mSme qu'Eisenstein, 16gu6 sur ce sxyet aux g^omfetres qa'un ioigme difficile k d^chiffrer. ' Sur les trois M<;moires prc^sent^s au concours, le premier a ^te 6cart6 comme insuffisant. ' Le deuxi6me suivait exactement la marche trac6e par M. Smith et donnait la demonstration de MB ^nonc^s ; celui des Commissaires qui a accept6 la tache d'en faire I'examen a pu, sur ces indices, deviner le nom de I'auteur. Peu importait, d'ailleurs, que le M6moire ffit de M. Smith ou inspire par INTRODUCTION. Ixxi le travail depuis longtemps livr^ au public par le savant professeur d'Oxford : il m^ritait incontestable- ment le prix. ' Un troisifeme M^moire r^solvait la question ; il 6tait difiBcile que deux g^omfetres assez habiles pour parcourir ce terrain ^leve, mais un pen etroit, ne s'y rencontrassent pas sur plus d'un point. Les m^thodes avaient de I'analogie, mais chaque M^moire portait la marque d'un esprit original et distingu6 ; tous deux etaient excellents et il semblait impossible de donner k I'un d'eux le second rang. ' Les deux Memoires seront publics, et I'Acad^mie se Micitera d'avoir donne k leurs savants auteurs, Tun k la fin, I'autre au debut de sa carrifere, I'occasion de montrer les ressources d'un esprit ingenieux et la preuve, inscrite k cbaque page, d'une science etendue et profonde.' These official remarks, which are supplementary to the report of the Commission, render justice to M. Minkowski, and offer a carefully framed defence of the Academy, but without admitting that the subject was proposed in ignorance of Professor Smith's work, or that the reporter was not aware of the existence of the paper of 1867 untU after the publication of the report. In a historical statement relating to the subject and award of the prize, drawn up a fortnight after the publication of the report, and in reply to adverse criticisms, a full avowal of all the circumstances might have been looked for. It is right to say that M. CamUle Jordan, the reporter, was not a member of the Academy when the subject was announced, and that it was orJy at the last moment that he was charged with the duty of reporting upon the three memoirs. It is much to be regretted that it should have been necessary to devote so much space to the matters connected with this memoir. A very brief notice would have sufficed if M. Hermite had communicated the existence of the paper of 1867 to the other members of the Commission, or if after the award he had given a brief account of the facts, or caused such an account to be given. But the only statement made was that of M. Bertrand, and it therefore became impossible to avoid details and quotations, as Professor Smith would not have been willing to send in a memoir for the competition except under the special circumstances of the case and in response to M. Hermite's suggestion. An Appendix at the end of the second volume contains four writings which, though not of the same original character as the papers themselves, neces- sarily find a place in a collected edition of Professor Smith's mathematical works. The last of the four is a portion of the Introduction to the collected edition of Clifford's Mathematical Papers, which was written in the summer of 1881. Only 80 much of this Introduction has been included as could be of interest to a reader who had not the book itself before him. A reference should be here added to a review by Professor Smith of Campbell and Garnett's Life of Professor Ixxii INTRODUCTION. Clerk Maxwell which appeared in the Academy for January, 1883 (vol. xxiii, pp. 19, 35). This review, being ahnost wholly biographical, is not reprinted. He contributed verbally to the meetings of the London Mathematical Society and British Association a number of papers, which unfortunately were never written out. The following is a list of the titles of these papers : London Mathematical Society. 1. Construction of the last point of intersection of a cubic curve by a curve of a superior order. March 26, 1868 (vol. ii, p. 61). 2. Geometrical note on the concomitants of a binary cubic. March 26, 1868 (vol. ii, p. 61). 3. Theory of certain systems of conies which present themselves in connexion with cubic curves. May 28, 1868 (vol. ii, p. 67). 4. On a problem in kinematics, and focal properties of skew surfaces. April 14, 1870 (vol. iii, p. 99). 6. On elliptic integrals. December 8, 1870 (vol. iii, p. 195). 6. On skew cubics. March 9, 1871 (vol. iii, p. 224). 7. On the partition of geometrical curves. February 10, 1876 (vol. vii, p. 90). 8. On the aspects of circles on a plane or on a sphere. April 13, 1876 (vol. vii, p. 172). 9. On some elliptic function properties. January 11, 1877 (vol. viii, p. 139). 10. On Eisenstein's Theorem. June 14, 1877 (vol. viii, p. 289). 11. Note relating to the theory of the division of the circle. April 11, 1878 (vol. ix, p. 102). 12. On a correction in Sohncke's tables. January 9, 1879 (vol. x, p. 44). 13. Upon a modular equation. January 9, and February 13, 1879 (vol. x, pp. 42 and 75). 14. Two geometrical notes relating to surfaces of the second order. March 13, 1879 (vol. x, p. 104). 15. Two geometrical notes. June 12, 1879 (vol. x, p. 167). 16. Geometrical notes (3). February 12, 1880 (vol. xi, p. 50). Britigh Association. 1. On a property of surfaces of the second order 2. On the large prime numbers calculated by Mr. Barrett Davis 3. On a construction for the ninth cubic point 4. On geometrical constructions involving imaginary data 6. On a property of the Hessian of a cubic surface . 6. On the circular transformation of Mbbius . 7. On modular equations 8. On singular solutions 9. On the eflFect of qnadric transformation on the singular points of a curve 10. On the modular curves 11. On quadric transformation 12. On inverse figures in geometry ..... 13. On a mathematical solution of a logical problem 14. On the distribution of circles on a sphere . 1866, p. 6. 1866, p. 6. 1868, p. 10. 1868, p. 10. 1868, p. 10. 1872, p. 24. 1873, p. 24. 1875, p. 21. 1875, p. 21. 1878, p. 463. 1878, p. 465. 1880, p. 476. 1880, p. 476. 1880, p. 476. Nottingham. Norwich. Brighton. Bradford. Bristol. Dublin. Swansea. INTEODUCTION. Ixxiii 15. Note on the skew surfaces of the third order . . , 1880, p. 482. Swansea. 16. On a kind of periodicity presented by some elliptic functions 1880, p. 482. „ 17. On the differential equations satisfied by the modular equa- tions 1881, p. 535. York. 18. On the equation of the multiplier in the theory of elliptic transformation . . . . . . . , 1881, p. 538. „ 19. On the linear relation between two quadratic surds . . 1881, p. 538. „ 20. On a property of a small geodesic triangle on any surfiice . 1881, p. 548. „ I have omitted a title from this list whenever I knew that the paper in question was published elsewhere. Thus a paper ' On Continued Fractions' which was communicated to the British Association in 1875 was afterwards published in the Messenger, and forms No. xxviii. of the present reprint. It is probable that the contents of a few others are included in the published papers. No doubt all the results upon which these communications were founded are contained in his note-books. With respect to the character of Professor Smith's mathematical writings a very noticeable feature is the arithmetical spirit that runs through the whole of his work. The years of study which produced the Report upon the Theory of Numbers exercised a lasting influence upon his mode of thought ; and his familiarity with the ideas and methods of the Higher Arithmetic continually shows itself in his treatment of Geometry and Elliptic Functions. In the latter subject the arithmetical tendency of his mind is especially evident in the point of view from which the theory of Transformation is always regarded. Another characteristic feature of his work is its completeness, both as regards attention to details and accuracy of demonstration. He had a very strong dislike to careless or slovenly work of any kind, and thought that it was nowhere so much out of place as in Pure Mathematics. He was ready enough to pass over the ground boldly and rapidly, without regard to ambiguities or details, when he was seeking after new theorems, or merely endeavouring to decide upon the tinith of generalizations or guesses ; but he was of opinion that a mathematician should refrain from publication until he had established his results by perfectly rigorous demonstration. He had no sympathy with those who were contented to give imperfect demonstrations, or to regard results as proved merely because they had satisfied themselves of their truth. No task is more irksome to a mathematician than that of working out in detail all the various particular cases of a theorem, when the novelty of the investigation by which it was discovered has long since worn off". The general result, too, of such examinations is to produce modifications and limitations which at the same time add to the cum- k Ixxiv INTRODUCTION. brousness of the demonstrations and detract from the simplicity of the theorems themselves. But he held that any slurring-over of difficulties or ambiguities was utterly repugnant to the nature of the subject, and that a mathematician was bound to spare no amount of labour that was requisite in order to give to his results the highest degree of precision of which they were susceptible. The comparatively slow rate of progress of the memoir on the Theta and Omega Functions was no doubt primarily due to the many other claims upon his time, but it was also attributable, in no slight degree, to the extreme care taken to avoid ambiguities of every kind, and to the attention bestowed upon the systematic examination of all the special cases of the general theorems. His natural love of precision in thought and expression was no doubt strengthened by his early study of the wTitings of Gauss, for whom he always felt the most unbounded admiration. The following notes, which he wrote for Mr. Tucker*, on the occasion of the celebration of the centenary of Gauss's birth, find a fitting place here, as they show, in his own words, not only his deep reverence for the great master of the Higher Arithmetic, but also the extreme importance that he attached to perfection of form in the presentation of mathematical results. If we except the great name of Newton (and the exception is one which Gauss himself would have been delighted to make) it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute rigorousness in demonstration, which the ancient Greeks themselves might have envied. It may be admitted, without any disparagement to the eminence of such great mathematicians as Euler and Cauchy, that they were 80 overwhelmed with the exuberant wealth of their own creations, and so fascinated by the interest attaching to the results at which they arrived, that they did not greatly care to expend their time in arranging their ideas in a strictly logical order, or even in establishing by irrefragable proof propo- sitions which they instinctively felt, and could almost see, to be true. With Gauss the case was other- wise. It may seem paradoxical, but it is probably nevertheless true, that it is precisely the effort after a logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity and unnecessary difficulty. The fact is that there is neither obscurity nor difficulty in his writings, as long as we read them in the submissive spirit in which an intelligent schoolboy is made to read his Euclid. Every assertion that is made is fully proved, and the assertions succeed one another in a perfectly just analogical order ; there is nothing so fur of which we can complain. But when we have finished the perusal, we soon begin to feel that our work is but begun, that we are still standing on the threshold of the temple, and that there is a secret which lies behind the veil and is as yet concealed from us. No vestige appears of the process by which the result itself was obtained, perhaps not even a trace of the considerations which suggested the successive steps of the demonstration. Gauss says more than once that, for brevity, he only gives the synthesis, and suppresses the analysis of his propositions. 'Pmtea sed matura' were the words with which he delighted to describe the character which he endeavoured to impress upon his mathematical writings. If, on the other hand, we turn to a memoir of Euier's there is a sort of fi-ee and luxuriant gracefulness about the whole performance, which tells of ' Carl Fiiedrich Gauss,' by R. Tucker. Nature, vol. xv, p. 537 (April 19, 1877). INTRODUCTION. Ixxv the quiet pleasure which Euler must have taken in each step of his work ; but we are conscious nevertheless that we are at an immense distance from the severe grandeur of design which is character- istic of all Gauss's greater efforts. The preceding criticism, if just, ought not to appear wholly trivial ; for though it is quite true that in any mathematical work the substance is immeasurably more important than the form, yet it cannot be doubted that many mathematical memoirs of our own time suffer greatly (if we may dare to say so) from a certain slovenliness in the mode of presentation ; and that (whatever may be the value of their contents) they are stamped with a character of slightness and perishableness, which contrasts strongly with the adamantine solidity and clear hard modelling, which (we may be sure) will keep the writings of Gauss from being forgotten long after the chief results and methods contained in them have been incorporated in treatises more easily read, and have come to form a part of the common patrimony of all working mathematicians. And we must never forget (what in an age so fertile of new mathematical conceptions as our own, we are only too apt to forget) that it is the business of mathematical science not only to discover new truths and new methods, but also to establish them, at whatever cost of time and labour, upon a basis of irrefragable reasoning. The fiaBriimTums mdavoXoymv has no more right to be listened to now than he had in the days of Aristotle ; but it must be owned that since the invention of the ' royal roads ' of analysis, defective modes of reasoning and of proof have had a chance of obtaining currency which they never had before. It is not the greatest, but it is perhaps not the least, of Gauss's claims to the admiration of mathematicians, that, while fully penetrated with a sense of the vastness of the science, he exacted the utmost rigorous- ness in every part of it, never passed over a difficulty as if it did not exist, and never accepted a theorem as true beyond the limits within which it could actually be demonstrated. These words certainly express the ideal which Professor Smith had always in his mind, and which has governed the character of his own work. In passing the papers through the press I have corrected all the misprints and errors that I detected, but no other alterations of any kind have been made in the text. I have added notes only in those cases where they seemed to be absolutely necessary. All additions, references, or notes which are not in the original papers are enclosed in square brackets [ ]. The correction of misprints or slips often involved matters of some delicacy, and occasioned frequent delays. There were also other difficulties connected with the papers that were printed from manuscript. The sheets containing the concluding portion of the Report on the Theory of Nimibers were passed through the press (during my absence abroad) by Professor Cay ley, by whom the index to the Report was made. Professor Cayley also kindly undertook the revision of the uncorrected portion of the Memoir on the Theta and Omega Functions. Professor Smith did not leave many separate mathematical manuscripts, most of his work being contained in note-books. These books, about forty in number, cover the whole period of his mathematical career. Some contain his early notes when making his first studies in Geometry, or reading the memoirs upon which the Report on the Theory of Numbers was founded ; others relate to his University lectures ; and rather more than a dozen are devoted to the k 2 bcxvi INTRODUCTION. records of original work, a very large portion of which has never been pub- lished. I have repeatedly examined the note-books relating to the subjects with which I was most familiar in hopes of being able to make extracts that could have been included in the present volumes. But in this I have been unsuccessful, for Professor Smith entered in these books not only the finished theorems which he had demonstrated, but also results which he had arrived at by rough explorations and inductions, as well as mere guesses sometimes ; and it is certain that he would have published nothing himself from these books without submitting it to the most careful examination and working out the demonstra- tions afresh. Under these circumstances it was decided, but with great reluctance, to confine the present work to the published writings, and make no attempt to give an account of the varied contents of the note-books. The editing of any considerable portion of the unpublished work would be a matter of great diflSculty, requiring much time and research, but it would not be so serious an undertaking to prepare separately for publication some of the investigations which he has left upon special subjects. In particular, it is very desirable that his researches relating to the decomposition of numbers into seven squares should be published ; and it would probably be found that the editing of this application of the general formulae has been greatly simplified by his own treat- ment (in the prize memoir) of the corresponding woi'k on the five-square problem. The principal subjects upon which he lectured in the University were Modem Geometry, Analytical Geometry, Theory of Numbers, Calculus of Varia- tions, and Differential Equations. With the exception of the Theory of Numbers, his lecture-notes on these subjects are very fragmentary; but full and accurate transcripts of the lectures themselves as delivered were kindly supplied by Mr. Lazai-us Fletcher, F.R.S., Mr. Thiselton Dyer, F.R.S., Mr. H. T. Gerrans, Mr. Walter Larden, the late Mr. Arthur Buchheim, and other pupils. As no other teaching on Modern Geometry was given in an English University, and as his lectures on this subject exercised great influence upon the direction of mathe- matical studies in Oxford, it was considered very important that they should be published. The editorship was undertaken by Mr. H. M. Taylor, Fellow of Trinity College, Cambridge, who after a cai-eful comparison of the lectures as delivered in different years wrote out for press a fair copy of what might be regarded as the standard form of the course. It was, however, finally decided to abandon the publication, partly because the same ground was more systematically covered by foreign treatises, and partly because the extent of the lectures was 80 limited (owing to the fact that students did not specialize in the subject) that INTBODUCTION, Ixx VU the volume would be scarcely adequate to form an independent treatise on so im- portant a branch of Mathematics. It may be mentioned that the courses delivered in various years differed very much from one another, and it would appear as if their nature and extent had to some degi-ee depended upon the audience. It is well known that Professor Smith intended to write an Introduction to the Theory of Numbers, and regret was frequently expressed to him that the work was still unpublished. Among his note-books there are several in which the elements of the subjects are very clearly and succinctly explained in methodically arranged paragraphs, and it cannot be doubted that these are successive editions of the commencement of such a work, in which he was striving after greater perfection. Other note-books contain carefully written articles which may have been intended as chapters in such a work. The completed portion of the treatise, however, is so small, only reaching to quadratic forms, that the idea at first entertained of publishing it separately as a fragment was ultimately given up. I hope that it will not be thought out of place for me to include in this Introduction a few reminiscences of my own with respect to Professor Smith, as he appeared to me, and to attempt a sketch, however slight, of his personality. In the eleven years that have elapsed since his death many of those to whom his presence was so familiar have passed away, and a new generation of mathe- maticians has arisen to whom he is but a name, so that the time seems to have already come when it is allowable to place on record matters which were once of common knowledge or might have seemed too ti-ivial for mention in print. I first saw Professor Smith at the British Association meeting at Nottingham in 1866, when he was one of the secretaries of Section A (Mathematics and Physics). I can perfectly remember his attitude and manner both when as secretary he read the papers of others, and when standing by the black- board, he explained, so simply and gracefully, the nature of his own communi- cations to the section. His tall handsome figure, his commanding presence, and the charm of his manners, stand out clearly before me, as I watched him then ; and in no essential respect was there any change in him between the first time I saw him and the last. At this meeting he spoke upon the average frequency of prime numbers, and I then for the first time heard of Legendre's approximate formula ^ 1 ..oo^» for the number of primes inferior to x, a result which interested log X- 1-08366 ^ ' Ixxviii INTRODUCTION. me intensely, although I little thought that it was subsequently to occupy so much of my own time *. I was introduced to him in the committee-room of the section by my father, and although I was not eighteen years of age, he welcomed me with as much cordiality as if I had been a fellow-mathematician of equal standing with himself. I was a shy and retiring schoolboy, but, in spite of the respect with which his knowledge inspired me, his kind and friendly manner at once placed me at my ease. I mention so particularly this experience of my own because it was very characteristic of his gentle and considerate nature. I am sure that no one was ever treated by him with less courtesy or attention on account of youth or junior standing : on the contrary, I believe that in such cases he instinctively and unconsciously showed even more consideration, I may perhaps mention that on this occasion he gave me the first separate reprint of a mathematical paper which I ever possessed : it was not a paper of his own, but one which had been given to him, and seeing me interested in it he told me I might have it, as he could procure another copy from the author. I did not see him again till the meeting of the British Association at Brighton in 1872, the year after that in which I took my degree at Cambridge. At that meeting he spoke upon the circular transformation of Mobius. I was then able for the first time to appreciate his wonderful power as an expositor of abstruse mathematics. His winning manners and graceful delivery charmed me as before, but I was even more struck with the skill with which he succeeded in giving, in the simplest language, a correct idea of complicated theories to those to whom they were entirely new. * My memory is quite distinct that this account was given as a ' Report on the Theory of Numbers,' and that he briefly explained to the section the nature of the subjects dealt with in the report. This impression is confirmed by the fact that among the sectional papers there is no title under which legeudre's formula could have been introduced ; for the paper ' On the large prime numbers calculated by Mr. Barrett Davis ' (which I also distinctly remember), was given on a different day and in a different room. (Mr. Barrett Davis had communicated a manuscript list of large prime numbers, and Professor Smith, in laying them l)efore the section, merely called attention to the fact that, as in the case of the smaller primes, they were sometimes clustered thickly together and sometimes widely separated.) It would therefore appear, almost with certainty, that Professor Smith had intended to write a seventh part of the report, which should relate to the frequency of primes and other asymptotic formulse in the Theory of Numbers. The early note-books written while the report was in preparation contain references to Legendre's law, and a r^surad of Lejeune Dirichlet's memoir on asymptotic formulae in the Berlin Abhandlungen for 1849. Professor Pliicker was present at this meeting, and exhibited some models of complexes to the section on the same morning as that on which Professor Smith spoke upon the subject of his Report. INTRODUCTION. Ixxix All the papers of which he gave any verbal account In public after this date were communicated either to the London Mathematical Society or to Section A of the British Association, and I believe I was present on every such occasion ; for I was a very regular attendant at the meetings of the Mathematical Society; and was one of the secretaries of Section A from 1871 to 1880 inclusive. I was also present at the meetings In 1881 and 1882. Several of these papers, like the one I have just referred to, related to subjects with which I was quite unfamiliar ; but I never failed to derive some benefit from his explanations or to feel a deeper interest In the theories of Pure Mathematics in consequence of what he had said. In general I do not readily gain an Insight Into new mathematical methods merely from verbal explanations, but his papers had a wholly exceptional eifect upon me in this respect. He had the gift of fixing the complete attention of his audience, and Imparting valuable knowledge, no matter how remote or technical the subject. Those who were present at the reading of any paper of his will know that there Is no exaggeration in this. Some mathematicians of our day have regarded the reading of a technical paper before a society as a mere formal prelimlnaiy to printing, which exists only as a survival from the past : but the beautiful mathematical expositions by which Professor Smith could gently lead on his audience Into the remote intricacies of a difficult subject, prove that It is possible even In Pure Mathematics to convey a true idea of highly technical researches without being technical at all. He always began at a point from which an ordinary mathematical listener could take up the thread, and, laying down the main lines of his subject in a series of simple and clear sentences, following each other In logical order, succeeded, apparently with the greatest ease, In placing his audience In possession of sufficient general knowledge to enable them to grasp the nature and scope of the new work that he was bringing before them. He spoke slowly, with a marked emphasis and a measured and almost rhythmical utterance, which were very distinctive and attractive. His language was always peculiarly fehcltous, both in formal expositions and In private conversation ; and the elegance of his style may be fairly judged by the papers printed In the Appendix, which, I think, those who knew him could scarcely read without fancying that they heard in them the cadence of his voice. Although dignified in words, manner, and beai'Ing, he was utterly free from any trace of formality : and Indeed no small part of the charm of his character was due to the way in which natiiral dignity was modified by sweetness of disposition and gaiety of heart. Even when explaining IxXX INTRODUCTION. the most abstract theories with the severest logical accuracy, his liveliness and wit would frequently peep out unexpectedly in parenthetical remarks. He was always in touch with his surroundings, but never more perfectly so than when addressing a mathematical audience, for his modesty and unselfishness rendered it impossible for him ever to weary others by allowing himself to be carried away by the interest which he felt himself in the researches he was explaining. On no single occasion was he ever dull or tedious, and his papers were always looked forward to with pleasure at the Mathematical Society and by the habituds of the mathematical Saturdays at the British Association meetings. The power to render advanced researches intelligible and interesting to a mixed audience is a rare gift ; and the only other bialliant expositor of mathematics whom I have ever heard was Clifford, whose style however differed widely in almost every respect from that of Professor Smith. Clifford spoke very rapidly and fluently, in cleverly-worded sentences that were often startling or paradoxical. The art with which he could invest familiar things with a new interest, or connect them with novel ideas, and the facility with which this was done, apparently on the spur of the moment, were truly surprising, but it seemed to me that the effect produced was greatly dependent upon the exact words which he used and upon his mode of delivery. In Professor Smith's expo- sitions there was never anything paradoxical or artificial. The explanations which he gave were perfectly matter-of-fact, his power being shown in the skill with which he held the sustained attention of his hearers as he proceeded from step to step. It should be mentioned that very few of his papers were produced quite spontaneously. Mr. Tucker, the seci-etary of the Mathematical Society, was always anxious to have several communications announced for each meeting, and if he had not received enough titles would write to those who were likely to have papers in progress or suitable matter for verbal communication to the Society. Professor Smith always responded wQlingly to such appeals, and would mention subjects upon which 'he could say something, if required.' In the same way, at the meet- ings of the British Association at which he was present, I always asked him for papers, and he would give me a list of subjects which he could bring before the section, sometimes offering me a choice and letting me select those which I pre- ferred. In making his verbal communications he generally placed one of his quarto note-books on the table, open at the place, and occasionally referred to it as he proceeded with his explanation. These quarto note-books in their greyish covers were well-known objects to all who attended mathematical meetings between INTRODUCTION. Ixxxi 1873 and 1883. After laying before the Mathematical Society the results of some researches of his own, probably carried out years before, great pressure would be brought to bear to induce him to write out an account which would be suitable for publication. This he did whenever he could find the time, but unfortunately many of his most interesting communications remained unwritten when death removed him. The communications to Section A were never intended to be pubUshed in the volumes of the Association. One he wrote out for me for the Messenger (No. xxviii, vol. ii.), and others which I had specially asked for had been promised to me for the same journal. The address which he delivered before the Mathematical Society on retiring from his two years of office as President in 1876 possesses so much mathematical interest that I felt justified in including it among the papers (No. xxxi, vol. ii.). I think it would be admitted without question that this was by far the most remarkable presidential address, both in substance and in mode of delivery, which has been made to the Society. I have been thus particular in trying to describe the characteristic features of his method of exposition, partly because for some years before his death there was no more conspicuous personal figure in English Mathematics, and paiiily because in the severe style of the papers themselves there is no trace of the bright and winning gaiety of manner with which their first introduction to a mathematical audience was so often adorned. I should despair of the possibility of myself conveying any adequate im- pression of Professor Smith's position in University and general society, but fortunately I am saved from the anxiety of any such attempt by the excellent article in the Spectator from the accomplished pen of the late Lord Bowen, which is reprinted on pp. xlvi-li. This tribute of affectionate appreciation, in which Professor Smith's character is delineated with perfect justice and delicacy, enables the reader to form a true idea of the unique place which he held in the larger world in which he moved, while his special claims as a mathematician were unknown to all except a few experts. His general attainments were so great and varied, and his personal and social qualities so brilliant, that his mathematical powers were completely overshadowed by other more conspicuous gifts. In an article in the Times, published the day after his death, it was truly said : ' It is probable that of the thousands of Englishmen who knew Henry Smith, scarcely one in a hundred ever thought of him as a mathematician at all. . . . He was a classical scholar of wide knowledge and exquisite taste, and there were few who talked to him on English, French, German, or Italian literature, who were 1 Ixxxii INTRODUCTION. not struck by his extensive knowledge, his capacious memory, and his sound and critical judgment.' It always seemed to me very strange that it should have been possible for him to have held so distinguished a position in the foremost rank of mathematicians without his eminence, or his devotion to the subject, becoming more widely recognized among his friends and colleagues. His official post in the University was that of Professor of Geometry, and it was of course well known that he was an accomplished mathematician of high reputation. But I am sure that very few even of his intimate friends were aware that in his own subjects he stood alone in England, and that his papers upon the Higher Arithmetic held a place among the most important productions of the century in abstract science. Even fewer still had any idea of the extent to which his heart and mind were engrossed by his mathematical researches. This want of recognition (if it may be so called) was no doubt partly due to his dis- inclination to speak of his own work except occasionally to those whom he knew to be interested in it, and his non-mathematical friends may be pardoned for not discovering an enthusiasm which showed itself so little ; in fact it cannot be doubted that he would have been spared much of the voluntary work which he so unselfishly undertook at the solicitation of others, if the depth of his devotion to his own subject had been generally known. But I think a truer explanation is to be found in the fact that, as his whole time and powers were apparently given up to other occupations, such as University work of all kinds and Royal Commissions, it could scarcely be supposed that he would be much more than a dis- tinguished amateur in so exacting a science. There was nothing that suggested the specialist in his actions or conversation ; and it is indeed truly remarkable that, in the midst of so many varied pursuits all requiring constant care and attention, he should have been able to carry out original work which can compare in extent and profundity with the researches of the ablest mathematicians, who have con- centrated their whole lives upon their special subjects. Except in vacations he seemed to have no time for mathematical investigation, and the amount that he accomplished was always a mystery to me until I learned that after a hard day's work, closing perhaps with a dinner party at which his lively wit and brOliant conversation had made him seem the gayest and the brightest of the circle, he would quietly settle down to work in his own room for some hours before going to bed. What he then wrote related probably to matters that had been more or less in his mind all day, and to which at intervals he had actively turned his thoughts, making a few stray notes perhaps on slips of paper. The last thing INTRODUCTION. Ixxxiii of all at night he would enter the results of the day's work or thoughts in his note book. Most of his mathematical work he did in his head, by sheer mental effort, and he scarcely ever committed an investigation to paper in any detail except when writing it out for publication. The notes which he made whUe thinking out a subject were often written on scraps of paper or backs of envelopes, which were destroyed as soon as he had made a definite advance which would allow of an entry in his notes. The fact that he used pen and paper so little, relying on his brain as it were, increased the mental strain of his mathematical production, so that, as a rule, when struggling with difficulties or exploring new fields, he did not work for long at a time. After an hour or two he would leave the subject as it were to grow of itself in the background and permeate his mind, whUe he was actively employed on something less exciting *. I may here mention that the high standard of completeness which he exacted from himself in his published writings, and which has been referred to on p. Ixxiv, added considerably to the effort with which his finished work was produced. The logical sequence of propositions, the absolute sufficiency of definitions, and the rigour of demonstrations, were all matters that exactly suited the quality of his mind ; but his mode of working did not readily adapt itself to the laborious classification of the separate cases of a general theorem, or other details requiring merely industry and attention. As his attention was not specially directed to mathematics until after his degree, he was in fact as regards its higher branches a self-made mathematician. It was during the long period of isolated study in which he familiarized himself with every formula in the greatest of the abstract Theories that his powers were developed and that his interest in mathematics grew into the almost passionate attachment of later years. Led on by pure fascination, under no pressure, but without either assistance or encouragement, he slowly and surely mastered everything that had been accomplished, and gained such an insight into the principles of the subject, and such a command over its methods, as could * In an article in the Fortnightly Review for May, 1883, I wrote : ' His victories were won by the hardest of intellectual conflicts, in which for the time his whole heart and soul and powers were entirely and absolutely absorbed. It was in his wide interests and sympathies, the pleasure of inter- course with others, and the love of all that was good and cultivated, that he found relief from these severe mental efforts. Had he not been gifted with a disposition that gave him the keenest sympathy with every human interest, that attracted him to society and endeared him to his friends, that gave him, in fact, his other noble life — the life the world knew— his fierce devotion to the subject he loved would have ended his days long since.' 1 2 Ixxxiv INTRODUCTION. only have resulted from so long and complete a self-devotion. But one un- fortunate result of his comparative isolation was that he allowed too much of his own work to accmnulate in manuscript, and that, the ' note ' of personal ambition (as Lord Bowen described it) being wanting in his character, and no external stimulus prompting him, he remained indifferent to the advantages of early publication, and was too little sensible of the difficulties that would stand in the way of preparing for the press any work which has been too long on hand. Thus, when he was forty years of age, besides the Report, he had published only one important memoir, although he was in possession of an immense amount of original work relating to Quadratic Forms, Geometry, and Elliptic Functions. The foundation of the London Mathematical Society in 1865 was an event which exercised a marked influence upon the subsequent course of all his work. He was a fairly regular attendant at the council and ordinary meetings, and there met other mathematicians who appreciated his unique knowledge, and urged him to bring papers before the Society. Wherever he was known he waa a persona grata, but nowhere more so than at the Society's rooms in Albemarle Street. During his presidency he communicated nine papers to the Society (besides the Address), only four of which however were written out. As time went on, and engagements and duties thickened upon him, he became more and more uneasy about the mass of work that lay unfinished in manu- script. In declining to undertake a fresh piece of work he wrote : ' I have twenty papers embedded in my note-books. I extricated and published seven last year,' He found it impossible to obtain the amount of consecutive leisure that was requisite to complete long and difficult investigations ; and he was continually distracted between the fascination of new work and the desire to publish portions of the old. He would often say, ' I must bind my sheaves ; ' and only a few days before his death he said to his sister, ' My mind is teeming with new ideas*.' His power of reading rapidly the mathematical writings of others, seizing the principles and grasping the methods as if by intuition, always struck me as very remarkable. Up to the last, and in spite of the scanty allowance of time that • Three months before his death, after the meeting at Cambridge for a memorial to the late Professor F. M. Balfour, refei-ring to the opinion expressed by one of the speaker., that a man's original ideas came to him before he was thirty, he said to me that in his own case he was certain that not only had his power of seeing and understanding increased without interruption all through his life, but that his thoughts and ideas and invention had undergone a corresponding progression and development \ INTRODUCTION. Ixxxv he could devote to Mathematics, he continued to read new mathematical litera- ture with the same ardour, and he never allowed the pursuit of his own work to prevent him from keeping abreast of what was being done by others, not only in his own departments of study, but also in other branches of the exact sciences. I cannot refrain from devoting one brief paragraph to recording his admir- able style and perfect taste when addressing a mixed scientific audience. Of this I recollect three remarkable instances : the first when he proposed the late Professor Tyndall as President of the British Association for the meeting at Belfast in 1874 ; the second when, in reply to Lord Grimthorpe, he spoke on the endowment of research, at a special meeting of the Royal Astronomical Society, in 1881 ; and the third at the Balfour Memorial meeting, which has just been alluded to. On the second occasion especially, I think that none who heard the speech are likely to forget the power and brilliance of the speaker. He spoke so lightly, and often with such whimsical disparagement of his own attainments and performances, that even those who were conversant with the nature of his published writings and the varied character of his pursuits were frequently surprised to find how well acquainted he was with matters and subjects which would not have been thought likely to be of special interest to him. This was the case also in Mathematics ; and I can remember my own astonishment when, long after I knew him well, I accidentally discovered how familiar he was with every page of Jacobi's Fundamenta Nova. From the way in which the subject of Elliptic Functions was treated in his writings, I had not suspected that the Fundamenta Nova would have possessed so much attraction for him. The following extracts from a letter addressed to the late Mr. Todhunter (in acknowledgment of some reprints of his papers) seem to be of sufiicient interest to deserve preservation : I have been also reading, and with great interest too, your ' Conflict of Studies.' I am afraid I am a shade less conservative than yourself. I have been led to entertain a somewhat higher impression of the value of erperimental science, at least when the pupil is made to experiment himself. I am perhaps a little more willing than yon are to consider favourably attempts to improve Euclid, though I have a great dread of the Association for the Improvement of Geometrical Teaching. Further (as I am a professor, and as there is nothing like leather), I am for having more professors, with more work, and more pay. But I so heartily agree with much, or rather with most of your book, that I should not have troubled you with this letter, if it were not that I cannot wholly subscribe to your estimate of the present state of Mathematics. All that we have, one may say, comes to us from Cambridge ; for Dublin has not of late quite kept up the promise she once gave. Further, I do not think that we have anything to blush for in a comparison with France ; but France is at the lowest ebb, is conscious that she is so, and is making great efforts to recover her lost place in Science. Ixxxvi INTRODUCTION. Again, in Mixed Mathematics, I do not know whom we need fear : Adams, Stokes, Maxwell, Tait, Thomeon will do to put against any list, even though it may contain Helraholtz and Clausius. But in Pure Mathematics I must siiy that I think we are beaten out of sight by Germany ; and I have always felt that the Quarterly Journal is a miserable spectacle, as compared with Crelle, or even Clebsch and Neumann. Cayley and Sylvester have had the lion's share of the modern Algebra (but even in Algebra the whole of the modern theory of equations, substitutions, &c., is French and German). But what has England done in Pure Geometry, in the Theory of Numbers, in the Integral Calculus 1 What a trifle the symbolic methods, which have been developed in England, are compared with such work as that of Riemann and Weierstrass ! But it is with the younger, or at least the less-known, people that I feel the difference most. Our English papers are so often quite free from anything really new, whereas a German takes care to know what is known before he begins to work, and besides generally takes care to work at some really im- portant problem, and not at Fome trifling expression for the co-ordinates of the focus. If I had room, I should vent my spleen (or perhaps my envy) by saying that I attribute the mis- chief to the business of problem-making : ninety per cent, of the good problems in Pure Mathematics that I see, are, if I mistake not, mere fragments of some great theory, of which the candidate is supposed to be ignorant. In the last paragraph but one he refers to the want of a sufficiently important object in the papers of many English mathematicians. This was a subject which was often in his mind, and I have heard him more than once express his regret that so many writers, instead of attacking recognized difficulties or those parts of their subject where real advances might be expected, should be content to occupy themselves with developments of a comparatively trifling character. In connexion with the reference to Cambridge problems, I may mention that on one occasion, when I was telling him about a proposal to abolish the order of merit in the Mathematical Tripos, he said that in his opinion a system which was successful in extracting a great amount of hard work from the students should not (in spite of many drawbacks) be lightly abandoned. My own friendship with Professor Smith arose in connexion with the interest I felt in some of the subjects in which he was an accomplished master, but it was not until he began to write the Introduction to the Theta Tables for me that I became intimate with him. The progress of this work naturally brought us into closer and more frequent contact. I used to meet him at the Mathematical and Astronomical Societies, often walking with him to the Athenaeum Club at the close of the meetings, and we had long mathematical conversations at Cambridge when he came to the dinners of the Ad Eundem Club. When the memoir on the Theta Functions in its final form was passing through the press, we both read the proof-sheets, and at the same time he was sending me the Notes on Elliptic Transformation for the Messenger: I also had occasion to consult him on several mathematical and other questions ; INTRODUCTION. Ixxxvii and all these causes combined to produce a rapid interchange of correspondence during the last two years of his life. It was not until I became really intimate with him that I had any idea of the intensity and earnestness of his devotion to Mathematics. Even among mathematicians he referred so gaily and with so light a heart to his own studies and pursuits that I have been almost startled to find, when alone with him, how engrossed he really was with mathematical researches, and how completely they possessed his mind and affections. He derived intense pleasure both from working at Mathematics and from the contemplation of its truths and processes ; and although he was undoubtedly anxious in the latter part of his life that what he had accomplished should not perish in his note-books, he seemed quite indifferent to the amount of recognition that was accorded to his published writings by his contemporaries * : in fact, the only word of impatience that, so far as I know, ever escaped him with reference to the slight attention that had been paid to his best work, was the sentence quoted in the private letter to myself on p. Ixvi. The last paper of which he gave a verbal account had for its title ' On a property of a small geodesic triangle on any surface ' (p. Ixxiii), and was com- municated to the Meeting of the British Association at York in 1881. The object of this note was to point out that if a, 6, c are the sides of a small geodesic triangle, then the correction to be applied to the formula a- = ¥ + c^ —26c cos A is —^ (Area)^ x curvature. I have no word to express the admiration and affection with which I re- garded him myself As regards his qualities and abilities, if I had not known him as I did it would have seemed to me incredible that such varied gifts and powers could be combined in the same person. All the assistance that I have ever received with respect to the direction of my own work, or the manner of conducting research, came from him, and I have never ceased to miss his advice and help : and more and more with each succeeding year. It will be long indeed before his place in Mathematics can be held by another ; but in the lives of those who were personally indebted to him the void can never be filled. * In communicating a paper to the Mathematical Society he once had occasion to refer to some results contained in one of his memoirs in the Philosophical Transactions, and he playfully apologized for having ' to quote from a paper which he had no reason to think that any one had ever looked at.' His indifference to pergonal prominence or display of any kind was frequently shown at the meetings of the British Association, for whenever there was any pressure upon the limited time of the section, he always waived his own claims in favour of those of others. Ixxxviii INTKODUCTION. It is always somewhat hazardous to quote from private letters (except for the sake of facts), as they so often give to strangers a very different impression from that conveyed to those who knew the writer personally. Still I am tempted to close this Introduction with a few extracts from letters which, though too trivial perhaps to deserve publication on their own account, are yet not without a certain interest in connexion with the published papers. All the extracts are from letters written to myself during the last two or three years of his life ; and most of them have been selected because they relate to the progress of the Introduction (or Memoir) on the Theta and Omega Functions and the Notes on Elliptic Transformation, with which he was occupied to the very last. Oxford, 2 November, 1880. I enclose the penultimate copy of the four fl-functions. The Society is reprinting its early numbers, and I have ordered fifty separate copies. There is an erratum in the note on p. 9, viz. it should be, I think, /3 = iZ—v not ^ = v—v'. This I have altered in the reprint. The trodden worm will turn; and I feel sure that even Cayley will admit any defender of suffixes to all the privileges which appertain to the status of a worm. I therefore, speaking as a worm, declare that I do not in the least care for suffixes, but that any one who does not admit that a double notation is, for certain purposes, imperatively required by the circumstances of the case, is not fit to be an annulated animal at all, but only a mere zoophyte. I will, seriously, quite as willingly write ^( > x\ , or ^ {fj., v ; a;), as ^^^ y {x) ; indeed to me it is a mere printer's question. But if I am told that 3^,, ^-j; .S's' 3* (however convenient as abbreviations), or again 0, H, ©j, Hj, are as handy for use in general formulae applying to all the four 0-functions, I am disposed to dissent. The Germans, I perceive, are great lovers of suffixes ; and I confess that when I try to do without them, I soon want another alphabet. Of course you are most welcome to do what you please with my paper : it will be much honoured by any use you may make of it. The ' Logic,' such as it is, you should have had long since, but that I sit seven hours a day, day after day, with our Commission. ... It is my birthday and I am feeling very old. The paper referred to is No. xvi (vol. i). I had accused him of exulting in the number and complication of the suffixes, and had said that the criticism of Professor Cayley (who disliked suffixes and avoided their use as much as possible) would be, ' Too many suffixes ! ' I was in the habit of giving the principal theorem of the paper in my lectures, and had asked for the separate copies, as the formulae were unsuited for writing on the black- board. I had also said that when, in printing my 'Lectures,' I came to the Theta Functions I wished to reproduce the whole of the paper just as it stood as a separate chapter. The 'Logic' was the paper whose title appears as No. 13 on p. Ixxii. It had been promised for the Messenger. INTRODUCTION. Ixxxix Oxford, 5 June, 1881. Best thanks ; only I have not time to express them. [He then refers in detail to some misprints.] It is very kind of you to take the ti'ouble you have done about a wretched little paper, of which the only interest, if any, is that it applies Liouville's theorem to a question of convergence. . . . When I sent you the manuscript of my paper I had almost asked you to print (at the end of it) Eiemann's proof of Abel's ' little theorem.' There would then have been a good tail to a poor little thing ; because Eiemann's proof is a model of what such a proof should be. (I notice Todhunter in his 'Laplace's Functions ' refers a little contemptuously to the ' little theorem ' — this designation is mine, not his ; — and in this he is quite wrong, as I think the trigonometrical series at once shows.) The paper referred to in this letter is No. xl (vol. ii.), ' On some dis- continuous series considered by Eiemann.' Ryde, 15 July, 1881. Alas ! I am not yet at the Elliptic Functions. For three weeks I was tied to my sofa in Oxford by a sprained thigh : and during that time I was exposed to continual interruptions, as in addition to the usual Oxford business at the end of term I had become (just at that time) executor of my dear friend EoUeston's will, and guardian of his children. . . Finding I could not be quiet enough to work at my Introduction to your Tables, I took up a very different bit of work, the Introduction to Clifford's Collected Mathematical Works. This is three-parts done and must be finished next week early : indeed it would have been done long ago except that thinking about it takes me into space of many dimensions, &c. I giudge the time I am giving to it because I can say nothing on the one hand fit for mathematicians to read, nor on the other fit for non-mathematicians. So I have to maunder a good deal, which is neither acceptable to me nor suitable to my ideas of the right way of honouring Clifford's memory. I long to be at the Elliptic Functions, I can tell you. If you think the Messenger would like a note of three pages on one or two points in Riemann'a ' Hypotheses which lie at the basis of Geometry ' (viz. on the only two results wliich he announces in formulae), the said Messenger would be niobt welcome. On another occasion, he said that this Introduction was inferior to the similar work which he had done in connexion with the writings of the late Professor Conington, and that it ' savoured of the sick couch on which it had been written.' Oxford, 12 December, 1881. I have stolen a few hours for the Elliptic Functions, chiefly to try and get my hand in again for work immediately after Christmas. (Till then I am liable to many interruptions.) I mutt rewrite the transformations of the second order ; I fear that nine of them must be given, viz. the nine which give different trausformations of the ellvptic functions. Oxford, 7 February, 1882. I have not seen you for a long time, and am afraid I am not likely to see you very soon. I am a close prisoner to my sofa with an inflamed vein in my thigh (gouty phlebitis, they call it). I hope I am beginning to get slowly better, but it will be a good bit of time before I am able to move about again. I have had to rewrite ' Transfoi-mations of the Second Order,' Art. 33, and while about it I have also made many changes in Art. 31 (' Linear Transformations of Elliptic Functions'). All this I could send to the printer if it were any good as yet for me to do to. I am not allowed to ni ^ DJTBODUCTION. work very much, and I find I can only do rather easy things. But I think I am up to doing what remains to be done with the Memoir. It is horrible to me to think you should take any more trouble over the thing. And so I hope yon will do no more than look through the proof-sheets very hastily indeed, if indeed you do as much as that. I have been preparing a little ' paperlet ' to show (1) that tlie coefl&cients o, b of the general elliptic transformation a? I + a^x* + a^a^+ ... y~M 1+6, aj'+a, »;*+.. . are rational in A* and \', not only in u and v as they appear in Jncobi and Cayley ; (2) that when \* is an equal root of the modular equation, they are not rational (in general) in ^ and >} (viz. in this case Cayley's system of equations at the beginning of his memoir admits of more than one solution); (3) giving a new (slightly new) process for determining them which shows that in all cases (even of equal roots) they are rational in A* and X^ and — (in the equal-root case, — need not be rational in ^ and X'). I had thought of bringing this to the Mathematical Society on Tliursday ; but finding such a journey out of my reach I am thinking of inflicting it on you for the Messenger. By the way, I will try and finish the little fragment of Logic for you. My difiiculty is that I cannot get upstairs to my study and no one can find my papers for me. Oxford, 22 February, 1882. I am putting several interesting little things together in the ' Notes on Transformation ' which I am writing for you. Another extract from this letter has been given on p. Ixvi. Oxford, 9 March, 1882. I am sorry to say that to-morrow I shall not be able to be at the Astronomical Society. I shall however probably venture np to London in order to go to the Meteorological Office, under a solemn promise to my doctors to be carried up and do^vn stairs and do nothing else. So you see I am getting on, and if I am only patient I may soon hope to be about again. I have just finished going over the revise of sheets 2, 3, 4; and am sending them to you at Cambridge. I am ashamed to have kept them so long. I find a few errata of my own, but none (I hope) to give much trouble to the printer. I am putting together several ' Notes on Transformation ' for you. The paper is getting rather larger than I expected, because I have found two or three new (to me) little things while lying on my sofa. Several of his friends were desirous that he should be nominated as the President of the British Association. The following is an extract from a reply to a letter of mine on this subject : Oxford, 14 March, 1882. I can tell yon in a very few words what I feel about the Presidency of tLe British Association ; indeed I do not know any one more likely to understand my feelings with regard to the matter than yourself. I should esteem the office a most horrid nuisance ; at the same time I know my duty better to the British Association, to the University here, and to myself, than to refuse it if it were ofiered to me. For the honour (which I know to be a great one) I cannot bring myself to care (perhaps this is INTRODUCTION. XCl owing to a temporary weariness of the world, induced by lying on a sofa) ; but on the other hand I have a great horror of the indolence which induces one to refuse a position because the duties of it are irk- some ; and I think Dante was quite right to put the man in hell ' che il grau rifiuto & ' (I forget who he was, and what he declined). What makes me say that the position would be an unmixed nuisance, is that I have (by this time), in the University and out of it, had my full share of the sort of work which calls my mind away from the subjects which interest me most, and I am very anxious (before the evening closes in) to concentrate myself as much as I can. If I had to be President of the British Association, the best work of a year would have to be given to my address, and that is much more than I can afford. It would certainly be a sad interruption to my plans of work, and I should have a per- petual sense of unreality about it. He afterwards said in conversation that the only scientific topics of general public interest upon which he could usefully discourse in a Presidential address were the motion of the atmosphere, the law of storms, &c. Folkestone, 13 April, 1882. I have been here for a fortnight, and can now limp about enough for purposes of business. I hope to meet you on Friday, and to have a few words on Mathematics with you then. A quotation from a letter written a few days afterwards has been given on p. Ixvii. The prize memoir was completed and sent off by the end of May. Oxford, 30 July, 1882. Have you returned from the United States ? and, if bo, when and where can we have a conference 1 I have been absolutely idle for thirty days at Koyat in Auvergne, and have returned, a good deal better, I hope ; but I am totally demoralized, and I feel as if I was too sleepy ever to do anything like a day's work again. However, I am now your slave, till I have accomplished my engagements with you (Introduction — that was — and Messenger). But my mental forces are in complete disarray, and you will have to use the whip severely to rally them. Do you see that Lindemann has covered himself with immortal renown by proving the transcen- dentality of tt 1 Of course, nine-tenths of the discovery is really Hermite's : but then Lindemann has the immense glory of having seen that Hermite's metliod could be applied to prove the transeendentality of r, when Hermite himself despaired of it. I have never examined Hermite's method closely, but taking his results for granted, Lindemann's reasoning seems all right. It is difficult not to envy, as well as admire, people who do such beautiful things : Lindemann's name is sure of a place in every history of mathematics hereafter "■. * Nine years before (May 31, 1873) he had written to me: 'I am much pleased in particular with the way in which you call attention to the question of arithmetical irrationality. So far as n- is concerned, I do not believe that any one has ever proved even so much as that n cannot be the root of an affected quadratic equation. And I always maintain that, until geometers have done this, the}' should not treat the problem of the rectification of the complete circumference as a demonstrated impossibility. Perhaps, however, the proof of the quadratic equation theorem may be obtained by Lambert's method. But this I have never tried.' Dr. Lindemann was the guest of Professor Smith (when I was so too) at the Oxford Commemoration in 1876. m 2 zoa INTRODUCTION. Ryde, 20 August, 1882, I have four of my notes nearly ready for you, and hope to finish them before I leave. They will make about eighty of my little pages ; will this fill a number for you 1 I have been led in Note II to your question about convergence of series like sin am u in powers of u. The onlv one jrivine any trouble is ; here the radius of convergence is the analytical modulus ^ o o J sin am M of K or iK' or K±iK', whichever of these four is least ; and the question is to find the values of A' for which each of these is least. I have put headings to the Memoir, but have not sent it off, having been absorbed, so far as I had time, in my ' Notes.' But I will send it before I leave. On August 10 I had gone through the first seven sheets of the Memoir •with him at the Athenaeum Club. Ryde, 23 August, 1882. Can you let me have a figure in the Messenger t Here it is *. . . . It is one of the modular curves of order 4 ; it divides the plane (as you see) into five regions. The least possible ' quarter periods ' of sin am u are, if A^ lies in 1, 2, . . . , 5 (i. e. if the extremity of the vector 1<^ lies in 1, 2, . . ., 5), 1 . ^, ^ iK' ; 2.^iK',K; Z.^iK',K±\iK'; i.\iK',K±iK'; 5 .K±iK',\iK',i)xe, ± sign being taken according as >f is below or above the axis. The absolutely least period is put first; of course K and A" are the rectilinear integrals, and least refers to absolute magnitude, i. e. to analytical modulus. Of course also there is a general theory relating to transformation to which this proposition belongs (it is in fact the theory of a problem which Jacobi touches on in the Fundamenta Nova, saying it is very difficult). A more complete account of these results is given on pp. 411-413 of vol. ii. The first portion of the manuscript of the ' Notes ' was given to me at the meeting of the British Association at Southampton on August 29. Margate, 8 September, 1882. I return the proof. I am heartily ashamed of the state it is in. . . . My excuse is that I pressed myself a little too much to deliver the manuscript to you at Southampton. I am very glad I did so, however, for I think that it would have taken a longer time if I had tried to revise it thoroughly in manuscript, even allowing for the time it will take the printer to go through it. I think I have now made it hang together in an intelligible way. I confess that till I wrote out the pages, which I sent you from Spottiswoode's, I had imagined that, when the modular equation has equal roots, the multiplier might be a root (square or cube) of a rational function of k^ and X'. But I found that what really happens is that the multiplier (when the roots are equal) still continues to be a rational function of k' and X', but is a function of A^ and X.^ with irrational coefficients, viz. the coefficients contain an imaginary quadratic surd such as v — «*> where m is a whole number ; whereas in all other cases the coefficients are rational numbers. I had said nothing to contradict this ; but some of my * The figure represented a symmetrical closed curve, consisting of four loops, each of which included the next smaller one, and having three double points on the axis of x. The region ' 1 ' was the interior of the smallest loop, the region ' 2 ' the space interior to the next loop but exterior to the smallest, the region ' 3 ' the space interior to the next and exterior to the second loop, and similarly for the region ' 4 ' ; the region ' 5 ' being the space exterior to the whole curve. INTRODUCTION. XCUl present alterations are made with a view to lead up to it. More of them however are made simply to make the meaning, and connexion, clearer. It all lies close to what is known, but I think it is full enough of new little things to make it fit for the Messenger. I send all that I have received from you so that it ends abruptly. I ought to have before me this portion when I revise the remainder. Correcting this has taken me two and a half days of (for me) hard work. I return at once to Note II : but would you not prefer to follow up Note I with something else, and let Note II take its chance by and bye 1 Notes I and II together would carry you nearly to the end of a third number ; and this would be dreary for your subscribers. Please send a card to say you have received this and have not gone mad with indignation at the state of the proof. London, 16 September, 1882. I enclose the revise. Of course I need not see another revise, and I should think you need not, as Metcalfe might well be trusted to make the corrections. On Monday morning you shall have the manuscript of the remainder of Note I. Of course I could not resist the temptation of re-scribbling it. Enough of Note II to fill up the September number, and more, shall, if I can possibly manage it, be in your hands on Monday morning also. London, 17 September, 1882. I enclose the remainder of Note I, rewritten and made as tidy as I can. As for Note II, a great part of it is nearly ready, but none of it quite. I will send you, very soon indeed, as much as you are likely to want^and more. I am sorry to tell you it will make more than a number and a half. Now this is intolerable, and I must divide it, for I will not take up three numbers running (even if you would let me, which, for your credit as an editor, I hope you would not). I think I can manage to divide it, though some of the beginning part is written solely with a view to the end. Till I get the September number of the Messenger safe in your hands I don't look at the Memoir : alas ! Oxford, 24 September, 1882. Here is some more copy for Metcalfe. It will take him a good bit on into the October number. But now the worst of it is, that a lot more of Note II remains — I think twenty-five slips at least — and this is after my cutting off all about the absolutely least periods (with the curly cue curves), which I now propose to make into Note III (when you have got over the surfeit occasioned by Notes I and II). So that you see Note II, if allowed to run on, will take up nearly all, perhaps quite all, the October number. I cannot divide it into two Notes, because it really has a unity of its own, and the arithmetic of Arts. 2 and 3 (esjiecially Art. 3) would be unmeaning (in a note on Elliptic Functions) without what follows. But there can be no objection to your dividing it in print, with a ' To be continued.' And this I should advise you to do. But I put myself wholly in your hands, and will do what you please. I think I could let you have the rest of Note II very soon. The Messenger must have as many lives as a cat, if it survives my Notes. Still I am prepared to maintain that the stuff in them is reasonably good, though by trying to be complete and exact I have become diffuse. Brockham, 29 September, 1882. I enclose the rest of Note II. There is not quite so much of it as it looks. Still I think it will run on pretty far into the October number. I have (as I said) left out the parts that would require a diagram or two. If the London Mathematical Society are in want of food at their first meeting, I could give them an account of tliese omitted portions, which are to be Note III (when you allow such a thing to appear). This would also give me an opportunity of saying briefly what Note II comes to ' when it comes to be fired.' xciv INTRODUCTION. The next thing that I shall do is to send you the revise of the Memoir, and to this I shall now • stick till it is douc ; I shall begin at it this very evening. If Metcalfe could, without putting himself out, send me the whole of Note II together, it would save time. But I have treated him abominably about Note I, and only hope that Note II will come out decently straight ; there really are some things in it worth a moment's attention, Brockham, 4 October, 1882. I return the proofs. This time they are very clear, and Mr. Metcalfe will not be able to reproach me. A couple of references to Gauss and to ray own Report have to be inserted. I send with the proof a little fragment which comes in after the end of the August number, and before the beginning of the slips now sent to me. I also return the copy for the October number. All I shall want will be a revise (in pages) of the September number, and that will enable me to coirect the part that comes out in October. I do hope, and I think it likely, that I shall not run quite to the end of the October number. I think I shall be more easily forgiven by your public, if they see that I really have come to an end, and that someone else is going ahead. . . . AH this iutei-ests me very much, because it turns on the theory of ' reduction ' as applied to doubly periodic functions, and seems to me to excuse the amount of space I have made you give to it in the Messenger. I have still to make out whatever I can about the course of the curves P; but I fear this will not be much. I shall try (whenever Note III comes into existence) to put all this stuff into it. So Note III will want figures. My hexagon, curiously enough, had already been considered by Dirichlet ; not, of course, in relation to Elliptic Functions, but in proving Gauss' famous theorems about the minimum value of a ternary definite quadratic form. The concluding paragraph relates to the limits of convergence of a series for arg sn x, about which I had consulted him. He took great interest in the question, and several letters were entirely devoted to it. The portion that was written of Note III, referred to in the last three letters, appears as No, X of the ' Notes ' (vol. ii, pp. 408-414), Abergavenny, 7 October, 1882. I enclose the revise ; I see there is one page over, to run on into the October number . . . Your remark as to the complexity of the result in the case of the value of the series for arg sn x, has made me begin to doubt whether I am really right in saying that one of the branches of the three-forked curve of discontinuity does really enter the circle of convergence. If it does the nature of things is a fool ; if it does not, I am a fool ; the latter hypothesis seems to me the more probable, and I gladly embrace it. Besides, I begin to see dimly a weak point in my demonstration. If only the curve can be coaxed into staying outside the circle the result will be the simplest possible, viz. that the series, when convergent, always gives the least value possible. It was on the 12th of October, that I went over all the manuscript of the ' Notes ' at the Athenaeum Club with him (p. Ixiv). Oxford, 31 October, 1882. At last I return the revise. I dare say it is fiill of blunders of mine, and is peppered over with priuter's errata, but I cannot find any more than I have marked. If the alteration of the note on p. 88 and the rearrangement on p. 89 are troublesome, it would not be ruinous if they were left alone. Item, on p. 96 the signs in lines 2 and 3 are not very wrong INTRODTJCTION. XCV as they stand, and might be left as they are ; I have now made them correspond exactly with the ' elementary matrices of Art. 3 ' ; as they stand, they do not. I am very sorry to have kept you waiting so long. I comfort myself by thinking that the number has not been expected with great impatience by any one. Oxford, 7 November, 1882. I am almost sorry you took the trouble of sending me a revise. "When I returned the proof I had intended to tell you that you might print it straight off. The small world that reads the Messenger will give an audible sigh of relief when they come to p. 99, aijd find there is no more of me. How- ever, you will have, within a year or so, to print Note III, and some figures with it. That done I absolve you from all further Notes on Elliptic Functions, and if I ever write them, I will inflict them on the London Mathematical Society, or the Quarterly Journal, or on the new Scandinavian journal, or on Sylvester's Journal, or on any one but you. London, 30 December, 1882. Do you happen to have a copy of the sheet pp. 423-431 that you could send me? I have two, but the printer could hardly make them out. I mean now to do nothing but proofs for a long time. I have the two sheets which follow those now printed off practically ready, and there is nothing to cause delay for a long time to come. Oxford, 20 January, 1883. I enclose four more sheets of the Memoir : the rest (as far as set up in 4to) will follow immediately. I am sorry to say that the first two of the sheets I send have had to undergo great alterations. This will not happen with any of the remaining sheets. I should be very glad to have, as soon as you can, the manuscript which is in your hands set up. For the next three or four months I can give a great deal of time to this work, and hope (D.V.) to bring it to a close. Oxford, 1 February, 1883. Best thanks for your letter. I cannot be at the R. A. S. to-morrow. ... I have returned to the printers four sheets of the Memoir for revise — but this includes the sheet which really has to be set up again, and made, I should think, into two. I find the stuff' (now that I have quite forgotten it) more intelligible and hanging together better than I supposed. I find many little slips of mine and some of the printers', but very few great blunders so far. It takes an enormous amount of time to go through it. I must write to you before the week is over about figures, and about completing, or rather shutting up, the whole thing: there are now 136 pp.; I think it will run to about 170, or a little over. Please regard this letter as not needing any answer. I shall see you on the ninth. These were the last words I was to hear from him. The 9th of February was the anniversary meeting of the Royal Astronomical Society, and I entered the Society's rooms expecting to meet him, and go over some of the sheets of the Memoir in the way that had become habitual to us ; but Mr. Stone, who had just arrived from Oxford, told me that he had died at seven o'clock that morning. J. W. L. GLAISHEK I ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. [Transactions of the Ashmolean Society, VoL IT, No. xxv. Read December 1, 1851.] J. HE principles of the Analytical Geometry introduced by Descartes effected a change in the nature of the science, the importance of which it is impossible to over-estimate. This change consisted in two things principally; first, in the creation of a wholly new system of Geometry by the side of the old syn- thetical methods ; and secondly, in the complete, though gradual, metamorphosis which these methods themselves underwent. For a long time, it is true, this second effect did not manifest itself. No one, for example, would say that the Pure Geometry of our great English mathematicians, of Newton, or Maclaurin, or Matthew Stewart, exhibits so essential a difference from the Geometry of the ancients, as that which strikes us in the works of far less original writers in the present day. But the change, though long delayed, appears now to be complete, and the geometrical methods, by long contact with analysis, seem to have acquired much of its spirit, and of its peculiar power and facility, and this without losing the intuitiveness proper to themselves. This has been the work of the last sixty or seventy years ; but it is historically interesting to observe that two of Descartes' contemporaries had anticipated the change, and had introduced methods into Pure Geometry wholly unlike its ancient resources. The theorems due to Desargues and Pascal are still primary in the geometrical theory of the conic sections, and the methods by which those results were obtained, so far as it is possible to judge, appear to have partaken fully of the generality of the results themselv^es. In particular, B 2 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. had the great work of Pascal upon Conic Sections been published, there can be no doubt that geometrical theories, which we now owe to the writers of the last half century, would have been in the possession of the world for a much longer period. Unfortunately all that now remains of Pascal's Conic Sections is comprised in a letter of Leibnitz to Pascal's executor, glvmg an account of the work, which had been submitted to him for his inspection, and earnestly recommending its immediate publication; and in a fragment of three or four pages in length, entitled ' Essals pour les coniques,' and written by Pascal when he was only sixteen years old. But even this fragment, though printed and circulated by Pascal himself, was lost for nearly a hundred years after his death, and the magnificent theorem contained in it remained fruitless till comparatively recent times. Nor has Desargues been more fortunate. His works are completely lost, and we are left to form our opinion of him from the isolated expressions of Pascal and Descartes, and from the virulent attacks of his enemies, of whom he appears to have had a great many. Had it not been for the ill-fortune of their works, Pascal and Desargues would have been the founders of modern Geometry; as it is, to Monge, before all others, this honour justly belongs. Himself a great master of Analytical Geometry — how little, for example, has been added to the general analytical theory of surfaces since his time — he yet seems to have been the first who fully felt of how great an extension the old Geometry was capable. He not only enriched it with a method and a body of doctrine, which has rendered thoroughly rational the relation of the science to the arts depending on it, but he also Infused into Geometry two qualities which had seemed peculiar to analysis — method in its processes and generality in its results. What gave, and what still gives, analysis so immense an advantage over Geometry, is, that when we have once expressed analytically a definition, say, or a theorem, the known laws of the combination of symbols enable us to transform that expression in a thousand different ways, and whenever we can Interpret any such transformation we have a new theorem. To take an instance, perhaps too simple, if we write a''' + y* = a* we express merely the common definition of the circle ; if we write y'^ = a? — x^ we express a theorem, deduced from the definition, and deduced, too, by changing the place of a single letter in an equation. If we add to this copious power of transformation, first, the generality of analysis, that is, Its power of expressing theorems, essentially the same, however they may differ accidentally, in one and the same formula, and, secondly, the faxjility with which relations dependent on the consideration of Infinity may be algebraically expressed and ON SOME OF THE METHODS AT PRESENT IN USE IN PUBE GEOMETRY. 3 transformed, we shall perhaps have stated the three principal prerogatives of analysis over the old methods of Pure Geometry. Now it would be hazardous to say that rational Geometry can at present compete with analysis in any one of these respects, or even that it can ever hope to do so, but it is not too much to say that it is in possession of extensive and powerful methods for the transformation of theorems ; that it has attained to a generality of expression unknown before ; and, finally, that it has learned to employ all the resources of the theory of infinitesimals, and to employ them with facility and elegance for the demonstration of theorems that seemed formerly to require the aid of analysis. Nay, further, whereas since the time of Monge Analytical Geometry has received three principal improvements, the introduction of symmetry, the method of triliteral or quadriliteral coordinates, and the method of tangential coordinates, it is a fact that for the two last we are indebted, in the first instance, to the rapid development of Pure Geometry, and to the efforts suc- cessfully made by analytical writers to reconquer the ground that seemed for the moment lost to their favourite branch of the science. Among the principal peculiarities by which the present Geometry dis- tinguishes itself from the old we must reckon the theory of transversals, the different theories of the transformation of figures, and the frequent use of the geometrical method of infinitesimals. The object of the present paper is very briefly to characterise the first two of these theories, and to illustrate the last by a few examples of its application to the theory of geodesic lines. But before doing so it will be well to allude to a question, very obscure in itself, but which puts the spirit of the new Geometry in a clear light, I mean the theory of imaginary quantities and imaginary figures. The analytical writers on Geometry have adopted one of two courses in this matter. They have either attempted to construct imaginary magnitudes geometrically, or else they have asserted the impossibility of constructing them, and have thence inferred the impossibility of getting any good at all out of them. With respect to the attempted constructions of imaginaries, it cannot be denied that they have been of great use ; but they have been of use not to Geometry, which they have conducted to no new results, but to analysis, which they have enriched by an important interpretation of symbolical expressions. In fact, the persons by whom these constructions were introduced were much more familiar with analysis than Geometry, and hence they were guided much rather by ideas of analytical than of geometrical continuity. If, for example, in the equation y^z=a'^- x^ we assign to x values beyond the limits + a, and proceed to construct B 2 4 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. the iraa^nary ordinates in a plane perpendicular to the plane of reference, we preserve it is true the algebraic continuity of the function, and we give an admissible interpretation of the symbol s/~^'' ^^^ what geometer can persuade himself that the equilateral hyperbola thus obtained stands in any but the most arbitrary relation to the circle ? The continuity of the circle requires that every line in its plane, without exception, should cut it twice, and that every point in its plane, without exception, should be the intersection of a pair of tangents to it ; does the equilateral hyperbola enable us to realise or to interpret either of these properties in the cases in which they become unmeaning ? Again, it is an obvious remark, though apparently either not made or not attended to, that through every imaginary point there passes one, and but one, real line, and that on every imaginary line there exists one, and but one, real point. Can the principle of perpendicularity fiimish any explanation of this fact ? Apparently not; and yet this is only one out of many cases that might be mentioned. Analytically considered, the theory is faultless ; but geometrically, it introduces discontinuity, it is inadequate to explain the phenomena, and (what is still worse in the eyes of a geometer) it is barren of results. Nor should we forget that till the constructions in question can be extended to loci in space, their use can never become general even in Plane Geometry, since it is frequently requisite to consider plane curves as sections of surfaces. For these reasons all attempts to construct imaginaries have been wholly abandoned in Pure Geometry; but, by asserting once for all the principle of continuity, as universally applicable to all the properties of figured space, geometers have succeeded, if not in explaining the nature of imaginaries, yet, at least, in de- riving from them great advantages. They consider it a consequence of the law of continuity, that if we once demonstrate a property for any figure in any one of its general states, and if we then suppose the figure to change its form, subject of course to the conditions in accordance with which it was fii^t traced, the property we have proved, though it may become unmeaning, can never become untrue, even if every point and every line, by means of which it was originally proved, should wholly disappear. In this way geometers are enabled not only to present theorems, in appearance the most dissimilar, as really identical (which in a scientific point of view is of immense importance), but also to make one easy demonstration serve where many disdimilar ones were before required. The practice of demonstrating real properties by means of imaginary ones was first introduced by Monge, who employed it, though tacitly and without ON SOME OF THE METHODS AT PEESENT IN USE IN PUEE GEOMETEY. 5 enunciating the general principle, in demonstrating what we should now call the properties of poles and polars in surfaces of the second order. The principle was first broadly stated, I believe, by Poncelet, in his ' Traits des Propridt^s Projectives.' It was somewhat unfavourably criticised by the Commission which was appointed by the Institute to report on that work, and which consisted of Cauchy, Arago, and Poisson. It seems however to have more than held its ground, and to be frequently appealed to by all the most eminent writers. A single example of its application must suffice. It has been shown very recently that if we draw two tangents to a conic from a point without it, and inscribe a circle touching the two tangents and the curve, the arc of the curve intercepted between the two tangents will be divided at its point of contact with the circle into two parts such that their difference shall be geometrically rectifiable. This theorem is of great importance : it enables us to construct with extreme simpUcity the principal formulae for the addition and multipli- cation of elliptic functions of the two first species, and its demonstration depends ultimately on two general properties of confocal conies, viz., that if from any point on a conic A we draw two tangents to a confocal B, the angle between these tangents is bisected by the tangent and normal to A ; and secondly, that if from two points on A we draw pairs of tangents to B, these four lines shall be tangents to one and the same circle. Now it is a known theorem, due to Quetelet, that if two spheres be inscribed in a cone, so as to touch a plane section, the two points of contact will be the two foci of the section. It is also known that if a surface of the second order be inscribed in another surface of the same order, the two sections determined on the two surfaces by any plane will have double contact, the chord of contact being the intersection of the plane of the sections and the plane of contact. We may therefore consider the foci of a conic as two evanescent sections of a sphere, that is, as two evanescent circles, having double contact with the conic. If we now observe that an evanescent circle degenerates into a pair of imaginary lines, we shall perceive that a system of confocal conies may be regarded as all inscribed in one and the same imaginary quadrilateral. That is to say, they may be regarded as possessing all properties incident to a system inscribed in a real quadrilateral, and therefore as forming a system of curves in tangential involution, and from this general property the two properties required to be proved are at once deducible, though since the demonstration at this point ceases to be imaginary the subsequent steps are here omitted. Did time permit it might be shown how from the same imaginary property 6 ON 80MB OP THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. of confocal conies all the known theorems respecting such a system might be easily deduced, the geometrical theory of involution enabling us to transform the imaginary relation in a multitude of ways, and with a facility that renders the process almost mechanical. Nay, more, with merely verbal, or at least very simple alterations, the same proofs would apply to spherical as well as to plane conies ; so that, for example, we might extend the constructions before alluded to for elUptic functions of the second order, so as to include those of the third also. If any one will give himself the pains to examine any one of those theories of modem Geometry in which frequent use is made of imaginaries — for example, Poncelet's theory of homological figures, or Chasles' construction for the semi-axes of an ellipsoid — he will rise perhaps with no clearer idea of imaginary magnitude than wheh he began, but he will probably be satisfied that this is one of those cases to which Dr. Woodhouse's remark applies with all its force, that a method which leads to true results must have its logic. In passing to speak of the theory of transversals, it is hardly necessary to observe that only a very few of its most characteristic features can be alluded to here. Under the term Theory of Transversals I mean to include (somewhat improperly, though not without precedent) all those methods for the investi- gation of the properties of curves and surfaces which rest upon metrical rather than descriptive relations, and which operate on the figure as it stands, and not on figures derived by transformation from it. Several of the isolated theories belonging to this head appear to bear a special character, from the simple fact of their having been first developed as parts of the theory of Conic Sections. This is the case, for example, with the principles of harmonic and anharmonic section, and especially of involution. But it is a mistake to suppose that this special character is more than accidental. In fact, the harmonic properties of all geometric curves have been known since the days of Cotes, and have in recent times acquired great interest from the discoveries of Poncelet, and from the still imperfect theory of polar curves. But the importance of the theory of harmonic section cannot be fiilly understood till we reflect that of all con- ceivable metrical relations among points on the same right line, not one can by possibility belong to the sphere of linear Geometry, except it belong to the class of harmonic properties. With the ruler alone we cannot bisect lines or draw parallels, still less take proportionals, but we can always determine harmonic means and construct harmonic progressions. This observation alone would sufilce to show that the harmonic relation, far from being special in its application to curves of the second order, must meet us at every turn in the ON SOME OF THE METHODS AT PEESENT IN USE IN PURE GEOMETRY. 7 science of space, and such in fact we find to be the case. For example, let a geometric curve be traced on a sheet of paper, and let it be required to assign its tangent at a given point, not singular, with the ruler alone. Let P be the given point ; through it draw four transversals, which will cut the curve in n — 1 points apiece. On each of these transversals take the harmonic centre of its n—1 points with respect to P, and consider the four points thus obtained as determining a conic passing through P. Pascal's theorem wUl now assign the tangent of this conic at P, and this tangent will be the tangent required. We may add that were the curve of the wth class, instead of the nth order, i. e. were it such that only n tangents could be drawn to it from a given point, and if we supposed that we were given, or could construct, the tangents passing through any point in the plane, we might, by a reciprocal construction, determiae with the ruler alone the point of contact on any tangent, supposed not to be a double tangent to the curve. This solution of the general problem of tangents, being purely linear, is so far simpler than any that has ever been deduced from analysis, and it is worth whUe to ask why, in this particular case, Geometry possesses an advantage over analysis. The answer plainly is, that the analytical method introduces elements foreign to the real question, videlicet, a pair of right Hn§s termed axes, and standing in a purely arbitrary relation to the curve and its tangent at the point P. When therefore we proceed to construct the tangent by means of its relation to these axes (by constructing its intercepts, for example), we lose sight for the moment of its immediate connection with the curve, and substitute for that immediate connection a mediate, and therefore a less simple relation. It wUl perhaps be found that similar observations apply to almost all those cases in which the Cartesian analysis is outstripped either by the present Geometry or by the newer methods of trUiteral and indeterminate coordinates. Similarly, by properly generalising the definition of involution, we can obtain without any trouble at all very many general properties of systems of curves and surfaces, some of which would not be without interest, and would admit of a multitude of corollaries. Thus, if four surfaces have their complete intersection common, the an- harmonic ratio of their four tangent planes is constant for every point on the line of intersection, and reciprocally, if any number of surfaces of the nth order be completely circumscribed by one and the same developable, the anharmonic ratio of the points determined by four of the lines of contact on any edge of the developable is constant for all the edges of the developable ; so that, in 8 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. particular, if the lines of contact be lines of curvature on three of the surfaces, they are lines of curvature upon all of them. But, after all, it is in the method of transversals, strictly so called, that we find the principal resources of geometrical inquiry respecting general loci. That method is comprised entirely in the general theorem known by the name of Camot, and in the means that have been devised for transforming and developing the equation supplied by it. Its importance in Pure Geometry is so unquestionable that a few remarks upon the place which it holds in the theory of curves, and on the geometrical considerations by which it may be established, cannot be out of place here. What renders at all possible a purely geometrical theory of curves, is precisely the introduction of considerations involving imaginary points and lines into the principles of the science. It is impossible to define geometrically a plane curve of the nth. order, except by saying that it is a curve such that every right line in its plane must of necessity cut it in neither more nor less than n points, and of course this definition is inadmissible so long as imaginaries are excluded from consideration. But this definition once admitted, it is found possible to construct a purely geometrical theory of curves. If, for instance, we wish to find the locus of a point subject to certain conditions, these conditions give us the means of determining how many points of the locus can lie on one line, and, since the continuity of the locus implies that all lines in its plane cut it in the same number of points, we can in general determine the order of any proposed locus ; and as soon as this is done, we are in a condition to inquire still further into its properties, to construct it by points, to determine its general form, &c. We will take one or two very simple examples of this a priori determination of the order of a locus. The first shall be Cotes' theorem already alluded to, namely, that if we take a fixed point P in the plane of a geometrical curve, and draw transversals through it, and then take the harmonic centres with respect to P of the n points in which every transversal cuts the curve, the locus of these centres for every position of the transversal shall be a right line. Now it is clear that the distance of the harmonic centre from the fixed point, being a symmetrical function of the distances of the n points of intersection from the same point, can have but one value for one position of the transversal, and therefore but one point of the locus can lie on each transversal through P. Unless therefore P be itself upon the locus, either as an ordinary or as a singular point, the locus is a right line. But that P is not on the locus, may readily be verified in the case before us, so that the theorem is proved. As ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 9 a second instance, we will take the following from solid geometry. If a body be anyhow in motion, it is required to prove that the tangents to the trajectories of all the points in it, which lie upon a given line, form at any instant a hyperbolic paraboloid. To show this, we observe that no two tangents can lie in the same plane (except in a particular case, when all of them lie in one plane), and that consequently the complete section of the surface, formed by any plane containing the given line, consists of that given line and of one tangent only ; so that the locus is a surface of the second order. That it is a paraboloid now follows from the fact that all the tangents are manifestly parallel to one and the same plane. It should be added, that precisely in the same way in which we define a curve of the nth. order by the number of its points which lie on a line, we define a curve of the nth class by the number of tangents (real or imaginary) that can be drawn to it from a given point, and this definition enables us to find envelopes, just as the former enabled us to find loci : an obvious example will sitffice. ' One side of a constant angle passes through a fixed point, and the vertex lies on a fixed line, the other side will envelope a parabola.' For here the fixed line is itself one of the tangents, and therefore from each point upon it two, and only two, tangents can be drawn to the envelope — the envelope is therefore a conic section — and since the tangent line can in one position remove to an infinite distance, it is a parabola. We see then that when a locus is investigated geometrically, the most general and the most important question we can ask respecting it is, in how many points it can be out by a right line, and the preceding examples may serve to show how this question may in many cases be answered, that is, how the loci occurring in particular problems may be brought under the general definition of geometric curves. The next step is, from the purely descriptive relation asserted in the definition, to deduce an equally general metrical one ; and this is exactly what is effected by Camot's theorem. Camot gives, in the ' G^ometrie de Position,' two demonstrations of his theorem, one analytical and one geometrical. The former is very simple and elegant, but the latter is unsatisfactory; and though there can be no real objection to rest a general geometrical theorem on an analytical proof, it was still a problem of some interest to show that geometry could dispense with this assistance. To do this, M. Poncelet first showed that Camot's theorem passed into Newton's by perspective, and then succeeded in demonstrating the latter by considerations analogous to those which we have been just employing for the theorem of c 10 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. Cotes. Newton's theorem is, that if we take a fixed axis of abscissas, and draw ordinates to it, the ratio of the continued product of the abscissas to the continued product of the ordinates is constant, at whatever point of the axis of abscissas the ordinate is drawn. To prove this, we need only take upon each ordinate a Une proportional to the value of the ratio for that ordinate, there is then no difficidty in establishing, first, that the locus of the extremities of these lines is a right line, and secondly, that it is a right line parallel to the axis of abscissas. And this is evidently equivalent to the required proof Camot's theorem, notwithstanding its simplicity, is not very easily enun- ciated. If we take a triangle ABC in the plane of a curve of the nth. order, and if its sides taken in order be cut in the 3n points, PiPi. .pn, <?i ?a • • ?». **i ^'a ••»*>» ; and if we denote by (Aq) the continued product of Aq^ Aq^ .... Aqn, we shall have, by the theorem, (Aq) . (Br) . (Cp) = (At) . (Bp) .{Cq). We see that this equation establishes a relation between the 3 n points, which must of necessity subsist in order that they may all lie on one and the same curve of the nth order, and that when all these points are given, except one, we are able to determine that remaining one. If, for example, all the branches of a curve be completely described excepting one, and if two points upon the remaining branch be given, the theorem enables us immediately to describe it, or at least to determine as many points as we please upon it. But the principal applications of the theorem depend mainly upon the facility with which the fundamental equation may be modified and transformed. These modifications are rendered possible, first by the absolutely arbitrary position of the triangle in the plane of the ciurpe, and secondly, by the facility with which evanescent segments may be eliminated from the equation, by introducing new transversals, and combining the equations supplied by them with the original equation. Thus, if we take one of the vertices of the triangle of transversals as A upon the curve, Camot's equation will contain an evanescent segment on either side ; and if we introduce a new transversal, passing through the ex- tremities of the evanescent segments, and cutting the third side in a point t, the equation connecting the segments determined by this line on the sides of the triangle will immediately eliminate the evanescent segments, and give an equation for determining the point t, that is, for determining the tangent at A. By continuing the same process we might with the utmost facility determine the position and magnitude of the circle of curvature at A, and with a little more trouble might extend the investigation to the case of smgular as well as ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. H ordinary points. But our present limits preclude the possibility of our pursuing this subject further. We will only add that it is possible so to transform Camot's equation as to render the relation given by it capable of linear con- struction, and that in this way an immense number of descriptive properties of geometric curves may be obtained. For instance, we might demonstrate in this way Cotes' theorem, or the linear construction before given for the tangent at any point, which was exhibited as a consequence of the harmonic property of curves. But we will confine ourselves to merely stating one very general property due to Poncelet, ' If the points determined on the curve by a sufficient number of transversals be given, it is possible to determine the intersections of the curve with any other transversal by means of curves depending on the intersections of right lines only.' We must dismiss with a stiU more imperfect notice the theory of the transformation of figures ; and this, not because the subject is less interesting, but because, both in this country and on the continent, it has attracted so much more attention than any other part of pure geometry, that it would be no easy task to give a summary view of the whole system, while at the same time it would be hard to present anything with respect to particular appli- cations that should have the interest of novelty. A few general remarks is all that will be attempted here. Figures may be transformed in two ways ; either directly, that is, into others of the same kind, or inversely into reciprocal ones. In the first case, to every point in the original figure a point corresponds in the derived, and a line to every line. In the second case, this relation is inverted, and a point of one figure corresponds to a line of the other, and vice versa. It follows from this, that to the points of a curve line of the nth. order there will correspond in the first case the points of a curve of the same order, but in the second case we shall have as the correlatives of the points of a curve in the primitive figure the tangent lines of a curve, no longer now of the nth. order, but of the nth class. Consequently, the descriptive properties of a figure derived directly will be precisely the same as those of its primitive, but the descriptive properties of a figure derived inversely will be reciprocal to those of its primitive. This will frequently enable us to extend a descriptive relation from a particular to a general state of a figure, and from a descriptive relation of one figure to deduce another belonging to a different figure. The use of such processes in discovering new theorems, or in establishing a connection between ones already known, is too obvious to be dwelt on. We also see that any method of transformation, which satisfies the single condition, that to c 2 12 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. the points of a right line there should correspond the points of a right line in the first case, and a system of right lines passing through a point in the second, will enable us to generalise or transform any purely descriptive property. And it is possible to invent an unlimited number of methods of transformation, which shall comply with this restriction, and which, so far, possess no advantage one over another. But by a proper selection of the methods to be employed it has been found that we are enabled to transform not only all descriptive relations, but very many metrical ones also ; in fact, all such as can be enun- ciated in a suflficiently general form. The two methods of projection and of reciprocal polars, for both of which geometry is mainly indebted to M. Poncelet, besides including in themselves almost all the methods for transformation that had previously been proposed, possess this last property in a pre-eminent degree. What is singular is, that though the principles upon which the two methods rest are so widely different, exactly the same class of metrical theorems which can be brought under the first are capable of being also transformed by the second. AH harmonic properties, and consequently the whole 'geometric de la rfegle,' all anharmonic and involutional relations, and, besides, all the general theorems of the theory of transversals, can be operated on by either of the two methods. Wherever it is applicable, the method of projection will enable us to make the proof of a general theorem depend upon its simplest cases, and on the other hand to explain and follow the modifications which a general principle undergoes in its application to particular instances, while the method of reciprocal polars imveils the singular duality which pervades so large a portion of the science of space, and which now finds its analytical expression in the method of tangential coordinates, but which at the time of M. Poncelet's invention had hardly been observed at all. It was of course known that to any triangle upon the surface of the sphere there always corresponded a second triangle, the angles and sides of which answered to the sides and angles of the first triangle, and were connected with them by an uniform and simple metrical relation. But this remark had never been generalised so as to extend to all spherical figures, stiU less had it been perceived that the property in question was so far from being confined to figures on a sphere, that it was only a particular case of a general property of a far more extensive class of figures. Now, of course, it is well ascertained, that it is impossible to assert any theorem respecting a figure on a sphere without, at the same time, asserting a different property of a different figure on the same sphere, and this whether the theorem be metrical or descriptive. Even any proposition respecting the ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 13 rectification of a spherical curve gives us at once a theorem respecting the quadrature of the supplementary curve, so that if we could find aU spherical quadratures we could rectify all spherical curves, and vice versa. We can even in this way obtain transformations of definite integrals ; for example, of elliptic functions of the third order. For if we express first the area of a spherical conic, and then the length of the arc of its supplemental conic, we shall obtain two elliptic functions of the third order, with difierent moduli and parameters, and the supplementary relation of the two figures will at once establish an equation between these two integrals. The duality, then, of spherical figures is absolute, and extends to every conceivable case ; but as soon as we pass to plane figures, or to figures in space, the case is difierent, and it is only in certain definite, though stiU very extensive classes of properties, that we find the principle manifesting itself, though this perhaps may be partly owing to the imperfection of our means of investigation. For it is certain that in many particular cases it is a matter of considerable difficulty to discover the reciprocal relations between theorems, even where it can be shown to exist. Take, for instance, the two theorems, ' A tangent to the interior of two similarly placed and concentric conies cuts off a constant area from the exterior conic;' and again, ' The sum of two tangents to an ellipse, which intersect on a confocal ellipse, diminished by the arc intercepted between them, is constant.' No one would have suspected, at first sight, that these two theorems are supplementary to one another, in exactly the same sense in which the word is understood in Spherical Trigonometry. But we should find, that if we were to imagine the two figures to become infinitely small, and to be placed upon a sphere, they would become supplementary, and the properties specified would follow the one from the other. In the same way many properties of the asymptotes of an hyperbola might be shown to be supplementary to the focal properties of an ellipse. From the constancy of the sum of the radii vectores in the ellipse, we might deduce the constancy of the triangle contained by the asymptotes and any tangent to the hyperbola ; and from the equality of the angles made by a tangent to the ellipse with the focal radii vectores, we might infer the known theorem, that the intercept determined by the asymptotes on any tangent to a h3rperboIa is bisected at the point of contact. These instances may serve to show how cautious we should be in inferring that theorems, which seem to give rise to no reciprocal property, are really incapable of assuming this double character. It is therefore quite conceivable that future discoveries in Geometry may render the application of the principle of duality to the 14 ON 80MB OP THE METHODS AT PRESENT IN USE IN PURE GEOMETKY. general properties of space quite as universal as it already is in the case of the sphere. But in the present state of the science it would be hard to name a case in which the existence of duality can be proved, and in which, nevertheless, it cannot be brought to light by the method of reciprocal polars. This method, therefore, still extends to the full extent of our present knowledge. It is now equalled, but it has not yet been surpassed in this respect, by the analytical method of tangential coordinates. In forming the polar reciprocal of any proposed plane figure, we replace every point by its polar, and every line by its pole, with respect to an auxiliary conic taken in the plane of the figure. M. Poncelet has himself observed, that it would be unfair to argue against the generality of his method on account of its reposing (as it thus is made to do) on a particular property of curves of the second order. In fact, provided we once assure ourselves that the trans- formation we are employing is capable of being applied to any proposed figure, it will seldom signify whether the principles upon which the transformation rests be general or not ; the only object is to obtain such a transformation as will enable us to transform the greatest number of metrical relations possible. It has been said, that any projective property may be transformed into a re- ciprocal one ; but other relations, not projective, can nevertheless be made to yield reciprocal properties, by employing a circle or a parabola as the trans- forming conic. These particular applications, though of inferior interest with respect to general Geometry, are of great importance in the case of curves and surfaces of the second order, and in some physical applications of pure geometry — for example, in Professor Mac Cullagh's theory of apsidals, and his demon- stration of Fresnel's construction of the wave surface. Poncelet's account of Reciprocal Polars is to be found in a memoir in the third volume of Crelle's Journal, but he has devoted a separate work (the Traits des Proprietds Projectives) to the use of central projection, that is to say, of perspective in geometry. Few works, perhaps, could be named more calculated to awaken a taste for Pure Geometry than this admirable treatise : though the greater part of it is occupied with applications to the theory of conic sections, the reader feels all along that the methods developed in it are perfectly general, and that it only needs the genius of the author to apply them with equal success in almost any investigation. It is so natural an idea to simplify a diagram by forming a perspective representation of it, that it is surprising it should not have been introduced long before into geometry, especially when we remember for how long a time the conic sections were studied only on the cone. ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 15 Perhaps no better proof can be given of the rationality of the projective method than that which is supphed by the fact, that it is to it mainly that we owe the introduction of triliteral coordinates, a modification of the conception of Descartes, which possesses, it is true, many undeniable advantages over the purely geometrical perspective of Poncelet, but which cannot be properly under- stood in its relation to space tUl it is regarded, if we may so express ourselves, as a translation of perspective into the language of analysis. In fact, when we express the equation of a curve in triliterals, we are merely putting it into a form in which it becomes common to all possible perspective representations of the curve we are considering ; we, as it were, divest the curve of all its non- projective properties, for the purpose of exhibiting in a more palpable and ex- phcit form those essential ones which still continue to characterise it. We can immediately determine whether any proposed metrical relation be projective or not, by merely examining whether it leads or does not lead to a relation involving the angles that are at any point subtended by the lines of the figure, and not involving the projecting lines. Applying this criterion, we should find, for example, that Camot's theorem is projective, or, on the other hand, that it is impossible for a quadrature or a rectification to be so. But it would not be easy to lay down any general formula for determining a priori what properties are projective and what are not. Such a determination, though practically of httle use, would theoretically be of the greatest value. However, the criterion we have given enables us now to see what was before observed, that every projective property can be made to yield a reciprocal one. For let the auxiliary conic be a circle, we shall have a relation between the angles contained by the rays drawn firom its centre to the points of the figure, and since this same relation will subsist between the angles formed by the polars of those points, the new figure will possess a property reciprocal to and derived from that of the old. Lastly, it may be remarked that the principle of perspective seems well calculated to form the basis of classification for geometric curves of any given order ; a splendid example of this is given by Newton's famous theorem, that all curves of the third order may be generated by the shadows of five of them. This surely is the first step towards a purely natural classification of these curves, and it is much to be regretted that Pliicker in his enumeration of curves of the third order, which is the most complete that has yet appeared, appears to have paid so little attention to the fundamental distinction between curves that can and that cannot be cut from the same cone, for there can be no doubt 16 ON SOME OF THE METHODS AT PKE8ENT IN_ USB IN PURE GEOMETRY. that it would have enabled him to construct his classification of the 219 varieties on a principle at once more general and more simple than that which he has adopted. We come last of all to the method of infinitesimals. Ever since the in- vention of the Differential Calculus many of its resources have been at the command of Pure Geometry; and yet, in the first instance, the effect of its introduction upon that part of mathematical science was anything but favour- able to it. For as soon as Newton and his immediate successors had passed away, the geometrical methods ceased (with rare exceptions) to be cultivated ; the attention of mathematicians was so engrossed by the brilliant successes of the new calculus, that the era which is the most remarkable of all in the history of analysis was almost wholly unproductive in Pure Geometry. But the school of Monge, who delighted in finding geometrical solutions for all kinds of problems, soon attempted to rival the Differential Calculus on its own ground. And in particular cases they not only obtained very simple proofs of known theorems, but succeeded in discovering new properties, to which analysis might not have guided them so easily. Two striking instances of this might be men- tioned, both from the works of Ch. Dupin. One is his celebrated theorem, that three series of surfaces which cut one another orthogonally cut one another in their lines of curvature. This he demonstrated by direct and purely geo- metrical considerations ; and yet, perhaps, his proof is not the most simple that might be given. The other is the proposition which is the most general yet obtained in Dioptrics, ' That if a system of rays possess the property of being all normal to some one and the same surface, they will still continue to possess it after any number of refractions or reflexions at surfaces of absolutely arbitrary form and position.' Mains had succeeded in showing that if a system of rays emanate from a point they wUl form after a first reflexion two series of develop- ables intersecting orthogonally, i.e. that they will all be normal to the same surface. He then proceeded to inquire whether they would continue to possess this property after a second reflexion, and, deceived by a slight error in his analysis, he concluded that they would not. But Dupin showed, that if the surface normal to the incident rays were imagined to envelope a system of spheres of variable radius, but having their centres on the reflecting surface, those spheres would determine a second envelope behind the reflecting surface, and that every reflected ray would be normal to this the second sheet of the complete envelope of the spheres, that is, that all the reflected rays would be normal to one and the same surface. In the case of refraction, we have only ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 17 to substitute for the single sphere a pair of concentric spheres, having their radii in the ratio of the refractive index to unity*. Many other instances, some of them of even greater interest, might be given ; but instead of doing so we will conclude this Paper with demonstrations, simpler perhaps than those usually given, of some of the principal theorems relating to geodesic lines ; first, upon surfaces in general, and then upon the elhpsoid. We wUl set out with the assumption, which is easily justified, that a plane drawn parallel to the tangent plane at any point P of a curved surface, and at an infinitely small distance from it, cuts the surface in a curve which for all points indefinitely near the point of contact assumes the form and properties of an evanescent conic section, having its centre upon the noraial at the point P. If we now consider a point Q situated on the circumference of the conic, it is plain that the normal to the conic at Q will be the orthogonal projection on its plane of the normal to the surface at the same point Q. We can therefore find the angle which the normal at Q makes with the normal section PQ, and this angle, divided by the arc PQ, is equal to the reciprocal of the radius of torsion of the geodesic line PQ, since the normal section is the osculating plane of that curve at P, and the normal to the surface at Q is its principal normal at that point. Transforming the expression thus obtained for the radius of torsion, we obtain the still simpler one, 1 1 1 where Pj, R.2 denote the principal radii of curvature, while pi, p^ are the radii of curvature of the normal sections tangent, and perpendicular to, the geodesic line PQ. Several consequences may be deduced from this formula, a few of which will be mentioned here. First, if a geodesic line be tangent to a line of curvature, its torsion is invariably suspended at the point of contact. If therefore a line, of curvature become a geodesic line it must at the same time become plane, since every point upon it wUl be a point of suspended torsion. Secondly, if two geodesic lines intersect at right angles, their torsions at the point of intersection are equal. Thirdly, it is possible to trace upon a given surface lines of maximum * For an analytical proof of this theorem see the first part of Sir William Hamilton's Essay on Systems of Rays, or Prof. Minding in Poggendorff, 1847, p. 268. 18 ON SOIrTE OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. geodesic torsion, i.e. curves such that at any point upon them the geodesic tangent to the curve will liave greater torsion than any geodesic line passing through the point. Two of these curves will pass through every point on the surface. They will be orthogonal trajectories upon one another, and will in- tersect the lines of curvature at angles of 45°. Fourthly, if a point be found on a curve surface such that it is a point of suspended torsion on every geodesic line passing through it, it must be an umbilic. Consequently, if a surface have all its geodesic lines plane it must be umbilical at every point. But Monge has shown that the sphere is the only surface possessing this property; therefore the sphere is the only surface all whose geodesic lines are plane curves. Let us now consider any curve S traced on a given surface A. If P be a point on S, and if we project S orthogonally on the tangent plane at P, the projected curve will pass through P, and will possess at that point a definite curvature. This curvature we shall term the tangential curvature of the curve S at the point P; it is obviously equivalent to the curvature of S multiplied by the cosine of the inclination of its osculating plane upon the tangent plane. If (S be a geodesic line, its tangential curvature will be zero, and its tangential projection wUl be inflected at P, so that any small arc in the immediate vicinity of P may be regarded as rectilinear. It hence appears, that when S is not a geodesic line its tangential curvature is equal to the angle between two consecutive geodesic tangents, divided by the arc intercepted between the points of contact ; or again, it is equal to an evanescent geodesic chord, divided by the square of the sagitta bisecting it perpendicularly. Now if ^' be a second surface such that A can be developed on it without disruption or duplication, the minimum property of geodesic lines shows us that every geodesic line on A will determine a geodesic line on A', and that consequently the tangential curvatures of corresponding curves are equal at corresponding points on the two surfaces. In particular, we see that if ^ be a surface developable on a plane, the tangential curvature of S is precisely the curvature of the plane curve into which S is transfonned, when A is developed on a plane. Or, when A is any surface whatever, if we imagine it to be circumscribed by a developable along S, since the tangential curvature will continue the same, whether we regard S as traced on the developable or on A, we may define the tangential curvature of S as the curvature of the plane curve into which S is transformed by the complanation of the developable cu'cumscribing A along S. ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 19 Gauss, in his celebrated memoir ' Disquisitiones generales circa superficies curvas,' has introduced one or two expressions into Geometry which it is con- venient to preserve. If we take a finite area upon a curved surface, bounded by any closed contour whatever, and if through any fixed point we draw parallels to the normals to the surface along the given contour, they will inter- cept a spherical area on a concentric sphere of radius unity, which is termed the ' spherical value ' of the given curved area. The spherical value divided by the true value is the ' integral curvature ' of the area ; and if instead of a finite area we take an evanescent one, including a given point P, the limiting value of the integral curvature is the curvature of the surface at P. By the ' integral curvature ' of a finite arc of a plane curve we understand (it is hardly necessary to observe) the angle between the extreme normals divided by the arc, and the angle itself may be called the ' circular value ' of the arc. The theorems which we shall now endeavour to establish geometrically are the following : I. The curvature of a surface at any point is equal to the product of the reciprocals of the radii of curvature. (For simplicity, we consider only surfaces doubly concave, but the demonstrations, mutatis mutandis, will apply to surfaces having their curvatures of opposite signs.) II. The spherical curve, which is supplementary on the auxiliary sphere, to the spherical value of any proposed area, is equivalent to the integral of the angle of tangential curvature extended over the whole contour of the area. III. Corresponding areas on surfaces developable upon one another have equal spherical values, and consequently equal 'integral curvatures.' I. If we take an evanescent rectangle dS contained by four lines of curvature, it is plain that if S<p, Sep' denote the angles subtended by two adjacent sides at their respective centres of curvature, we shall have the equation dS = RR' S(}>S(f>'. But i{ dQ he the spherical value of dS, we also have dQ = S(bS(b'. Therefore j^ = j^y^,, and the truth of the result is inde- pendent of the peculiar form we have assigned to the element dS. For, whatever form we assign to that evanescent element, we can always imagine it made up of an infinite number of such rectangles, for every one of which the product jrpi will retain the same value within an infinitesimal, so that if da-, da-' etc. be the little rectangles, dw, dw' etc. their spherical values, we shall always find dS = ^da, dQ = '^dw ; D 2 20 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. and, in the limit, when dS is itself rendered evanescent, 1 du dto' dQ , fl = . . . = ~j-^, as beiore. RB^ da- d</ '" dS' II. Let S represent the contour of the given area, and 2 the contour of the corresponding spherical area. The spherical curve supplementary to 2, which we will term 2', is the envelope of great circles having their poles on 2, and, consequently, having their planes parallel to the tangent planes of the given surface along S. The consecutive intersections of the planes of the great circles determine the sides of the cone subtended by 2' at the centre of the sphere, and, in like manner, the tangent planes of the surface determine the arStes of the developable circumscribing the surface along S. Therefore the sides of the cone are respectively parallel to the ardtes of the developable, so that 2', which is obviously equal to the sum of the angles subtended by its elements at the centre of the sphere, is equal to the sum of the angles con- tained by the consecutive ardtes of the developable. Observing that every ardte intersects S once, and once only, we see that, after complanation, the sum of the consecutive angles will be precisely equal to the angle contained by the two extreme aretes, that is, to the angle contained by the two extreme normals, or, finally, to the integral of the angle of tangential curvature extended over S. A particular case of this theorem deserves special attention. Let S, instead of a continuous curve, form a polygon composed of geodesic lines. The theorem will evidently still subsist, only the quantity we have designated as the integral of the angle of tangential curvature will be simply replaced by the sum of the external angles of the polygon. We have therefore this theorem given by Gauss : ' The excess of the angles of any geodesic polygon, above the sum of the angles of a plane polygon of the same number of sides, is equal to the spherical value of the area of the polygon.' This property will serve, in its turn, to establish another of Gauss' pro- positions : ' If from any point O two geodesic lines OF, OF' be drawn containing an evanescent angle «o, and if OP = p, FF' perpendicular to OF = F, the quantity F will satisfy the difierential equation of the second order, dip ^_ dp' "^ RR'-^' Let Q denote the spherical value of the triangle OFF. Then, since ~ — l-Q — ft)=0, but also P = |Oco— / Qdp. Jo ON SOME OF THE METHODS AT PRESENT IN USE IN PTJBE GEOMETRY. 21 dP = dp cos OP'P, we find OP'P = ^ - ->— ; and therefore the preceding theorem 2 (t^ gives at once -^ — h i2 — &> = ; or, differentiating, —j-^ + -7— = 0. But, by theo- dQ P d^P P rem I, ^— = j^jt), and, substituting, -j-^ + ^^^ = 0. ttjO -flit/ (X,0 -flit We may also observe that this demonstration supplies us with a first and second integral of Gauss' equation. In fact, we have not only dJ^ dp The first arbitrary constant is w, the second has been put equal to zero in the inferior limit of the definite iritegral in the expression of P. In this form the equation suggests some interesting remarks, which our limits compel us to omit. III. The integral of the angle of tangential curvature, extended over any arc of a curve, is constant for all developments of the surface on which the curve is traced. For since each element of the integral is an angle contained by two consecutive geodesic lines, it is apparent that neither the number nor the magnitude of the elements wUl be affected by the transformation, and con- sequently the sum wUl remain unaltered. If the arc become a closed contour, it will foUow that not only this mtegral of contingence, but also the quantity supplementary to it, that is, the spherical value of the area, wUl remain constant for all developments of the surface. In particular, if we consider an evanescent area we shall find -^^ = -r^ • But since dil and dS are both constant, it follows that -777^ wUl be so too. iiii This gives us the theorem which Gauss has demonstrated by a singularly beautiful analysis : ' If two surfaces be developable one upon another, the product of the principal radii of curvature is the same for any two corresponding points.' The propositions we have been considering are of great importance in the theory of surfaces. The properties of geodesic lines, or more generally those properties of a surface which remain unchanged so long as the geodesic distances of its points remain unaltered, are doubtless as yet but very imperfectly known, notwithstanding the attention bestowed on them since the publication of Gauss' memoir. The subject is one of great difiiculty, as those who have tried it well know, but it is at the same time of great interest, as it is certain that any 22 ON SOME OF THE METHODS AT PRESEXT IX USE IN PURE GEOMETRY. general results obtained here would find frequent and useful application. Recently, too, the properties discovered by Mr. Roberts on the geodesic lines of the ellipsoid have attracted increased attention to the general question, and have themselves furnished fresh examples of the resources of Pure Geometry; for though Mr. Roberts' results were obtained in the first instance analytically, the geometrical proof of them since given is so direct that one is almost sur- prised that they were not discovered sooner. As it is possible to exhibit this proof in a very simple form, and one which shows clearly its connection with the general theory of surfaces, we will allow it to find a place here. We have to show that the sum of two geodesic lines drawn from the umbilics of an ellipsoid to any point on a line of curvature, including the two umbilics, is constant ; or, which comes to the same thing, that the angles made by the two geodesic radii vectores with the line of curvature are equal. Now it is a well-known property of confocal surfaces that the cones which envelope them from any point in space are themselves confocal, and consequently orthogonal, so that from whatever point in space two confocal surfaces be viewed their apparent contours will intersect at right angles. If we compare this property with the theory given by Monge, of the ' Surface of Centres ' of any given surface, we shall perceive that any two confocal surfaces may be regarded as forming the two sheets of the surface of centres of some one and the same transcendental surface, and that every geodesic line existing on either confocal surface, and touching their common intersection, will be the cuspidal line of a developable circumscribing the second confocal. This pro- position is the geometrical expression of Liouville's equation, m^ cos'i + v^ sin^i = p^, or Joachimsthal's, PD — const. As a particular case, we observe that every tangent to an umbilical geodesic line on an ellipsoid wiU pass through the focal hyperbola, so that the two tangents to the two geodesic radii vectores of any point on a line of curvature will be sides of the cone which from that point envelopes the focal hyperbola. But since these two sides lie in a principal plane of the cone they will make equal angles with the principal axes of the cone, that is, with the tangent and normal to the line of curvature. The same principles would serve to demonstrate the other theorems dis- covered by Mr. Roberts ; but the proof of this one may be enough to show, that though any property of space is doubtless discoverable by analysis, yet it sometimes happens that in particular cases it is more convenient to lay aside for a moment our analytical formulae, and consider the questions that arise in some more special but less artificial manner. This is especially requisite when ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 23 the subject of inquiry is one about which little is as yet known. For here we cannot be sure a priori that the analysis of coordinates is the most natural method, and therefore the surest to lead to results. In such cases we find ourselves obliged to adopt, though it be but tentatively, a more direct study of the circumstances of the case in preference to a method which is apt at times to conceal from us in a singular manner the real grounds of the results with which it supplies us. In the words of M. Poinsot, ' Rien ne nous dispense d'^tudier les choses en elles-memes, et de nous bien rendre compte des id^es qui font I'objet de nos speculations. Si le calcul seul pent quelquefois nous offrir une vdrite nouvelle, U faut songer que cette verite etant independante des methodes ou des artifices qui ont pu nous y conduire, il existe certainement quelque demonstration simple qui pourrait la porter a I'^vidence ; ce qui doit etre le grand objet et le dernier r^sultat de la science mathdmatique.' 24 ON SOME OF THE METHODS AT PRESENT IN USE IN PXTRE GEOMETEY. [The following abstract of the preceding paper was published in the Proceedings of the Ashmolean Society, Vol. H. pp. 305, 306.] The object of this paper was to show how, during the course of the last fifty years, the geometrical methods had acquired that generality and facility which had been, at an earlier period, regarded as exclusively characteristic of analysis. This rapid development of the resources of Pure Geometry was illustrated partly by general observations on the nature of some of its principal theories, and partly by a series of more particular examples. The use of imaginary magnitudes in Geometry was especially dwelt on, and it was shown how by their aid we may sometimes comprehend in one and the same statement theorems at first sight widely different, and exhibit them as expressions of some one and the same general principle, assuming a different form, under different accidental circumstances. Other illustrations were taken from the application of the theory of transversals to the investi- gation of the properties of algebraic curves, and, among other conclusions, a linear solution of the direct and inverse problem of tangents (analogous to that given by M. Poncelet) was deduced from the harmonic properties of such curves. The various methods for the transformation of figures, whether into other of the same kind, or into reciprocal ones, were also alluded to ; and the law of Geometric Duality, which manifests itself in these transformations, was commented on, and an attempt made to fix the limits within which it is applicable. In order similarly to exemplify the use of infinitesimals in Pure Geometry, some applications were made of this method to the theory of curved surfaces, and outline demonstrations of some of the results of Gauss on geodesic lines, and the mutual developability of surfaces, were given without the aid of coordinate Geometry. Lastly, a proof was proposed of Mr. Roberts' theorems respecting the geodesic lines of ellipsoid, in which those results were exhibited as immediate corollaries from two well-known theorems of Pure Geometry, due to Monge and Jacobi. II. ON SOME GEOMETRICAL CONSTEUCTIONS. [Cambridge and Dublin Mathematical Journal, vol. vii. pp. 118-126; May, 1852.] IF a geometrical curve be completely traced on a sheet of paper, the principles of the Theory of Transversals enable us to assign its tangent line and radius of curvature at any point, without supposing its equation known, and without employing any operations excluded from the sphere of elementary geometry. The construction given by M. Chasles for this purpose is the following. Let TO be the point on the curve, M any point assumed in the plane, mp, mq, two transversals, and MP, MQ, two parallels to them. Let also p, q, P, Q, denote the continued products of the segments on mp, mq, MP, MQ, respect- ively, excepting the evanescent segments on mp, mq. Then if we take on P Q mp, mq, two lines respectively proportional to — , — , the line joining their extremities shall be parallel to the tangent at m. To find the osculating circle, let t denote the continued product of the segments on the tangent at m, except- ing the two evanescent segments, T the continued product of the segments on MT drawn parallel to mt ; then, if on any transversal mp we take toc equal to -f- j}i the point c shall lie on the osculating circle. For the diameter of this T n circle we have the expression — ^ ; n, N denoting products of segments on the normal and on a parallel to it. If the point be a double point, the preceding constructions fail ; but by slightly modifying them, we may determine the two tangents and two radii E 26 ON SOME GEOMETRICAL CONSTRUCTIONS. of curvature, If the point be nodal : or, if it be conjugate, we can assign the elements of an ellipse, whose imaginary asymptotes shall be the imaginary tangents in question ; that is to say, an ellipse concentric, similar, and similarly placed, with the evanescent conic formed by the conjugate point. In this case the two radii of curvature are in general imaginary and therefore cannot be constructed : but, since they are conjugate imaginary magnitudes, any rational symmetrical functions of the two (for example, the rectangle under them, or their harmonic or arithmetic mean), may readily be determined. The process to be employed is as follows. Take three transversals mp, mq, mr, and three parallels to them MP, MQ, MR, and let MR cut mp, mq'u). A, B, and the two tangents in 6^ 6^ We shall have Ae,.Ae, = (mAy-y-^, B6,.Bd, = {mBy--^-^- Now if the point m be conjugate, the products AO^, A62, BOi, Bd^, are essentially positive ; and if it be nodal we can ensure their being so, by taking the three transversals mp, mq, mr, in one and the same pair of vertically opposite angles. Hence, if we put p n P n irn,Ay~-^=a\ {mBf^.^^h\ the lines a and & can always be constructed. Therefore to determine Q^, B^, describe two circles round A and B, with radii a and 6, respectively. The radical axis of these circles will Intersect MR at the middle point of 0„ O^; and any circle of the system orthogonal to the two circles {A) and {B), {i.e. any circle having its centre on the radical axis and its radius equal to the tangential distance of its centre from either of those two circles), will intersect MR in Q^, B^. If {A) and {B) intersect in real points, their radical axis is instantly found ; but in this case B^ and B.^ are always imaginary. Let s^, s^ be the points in which the radical axis is cut by any one of the orthogonal circles ; on mr take too' a mean proportional between os-^ and os^ (i.e. equal to the tangential distance of from any one of the orthogonal circles): the ellipse having its centre at m, and mo, mo' for semi-conjugate diameters, will have the two imaginary tangents for its asymptotes. If (A) and (B) touch, the points B^, B^ coincide in 0, and the double point becomes a cusp, having mo for its tangent. Lastly, if (A) and (B) intersect in imaginary points, the radical axis, though not immediately given, can always be determined by the ruler alone, and in this case, B^, B^ being always real, the tangents m0„ mB^ can be directly constructed. The direction of the tangents once ascertained, the radii of curvature may ON SOME GEOMETRICAL CONSTRUCTIONS. 27 be immediately found. In fact, if we denote by Cj, Ca the chords intercepted on mp by the two circles, and by Qi, 0^ the two points in which the tangents are cut by MP, parallel to mp, we have 01^2 c, = — ^ • — P 6162 T2 Co = — TT • — • ^ P mdi ti P' "' m02 k and by making mp coincide successively with the two normals, we get the values of the two diameters of curvature. Or we may first determine one, and then obtain the second by the proportion, which is easily demonstrated, -III : Jta . . — : — • If the point be triple the determination of the directions of its tangents, which in analysis depends on the solution of a cubic equation, is not in general possible by the intersections of right lines and circles. The problem in its simplest form is this : Given three points on a right line ABC, and given the products Aei.Ae^.AOa, Bdi.Be^.BOs, Ce^. Ce^. Ce^, find e^, e^, e^. But whatever the order of the point, if the direction of its tangents be once known, the construction of its radii of curvature is very easy. If, for example, the order of the point be r, the chord determined on mp, by the circle tangent to mOi, is readily seen to be T,p d,e2.e,e,...e,0r given by the equation ^'^Xp~ {md^y-^ ' which chord is therefore imaginary for an imaginary tangent, as it ought to be. Returning to the case of double points, we see from the formula mdi' ti P' that if the two tangents coincide, the osculating circles become simultaneously evanescent, except a fourth segment on the tangent become evanescent also, that is, except the tangent cut the curve in four coincident points at m. In this case the point m is not cuspidal, but is a point of osculation, and possesses two radii of curvature, for which we proceed to give a graphical construction. If T n Di, D2 be the two diameters, we find readUy enough DiDi= —-•^•, but the theorem of Newton's, which has hitherto guided us, is perhaps insufficient immediately to furnish a second relation. Such a relation, however, may be obtained by the following considerations. It is well known that the polar conic of a point of inflexion breaks up into two lines : one of these is the tangent at the point of inflexion, the other will be found to be the locus of the harmonic £ 2 28 ON SOME GEOMETKICAL CONSTRTJCTIONS. centres of the ?» - 1 points in which the curve is cut by a transversal through the point. Exactly in the same way the polar curve of the third order at a point m of the nature here considered, resolves itself into the tangent line and into a conic section ; this conic touches the tangent at m, and is the locus of harmonic centres of the n - 2 points in which the curve is cut by a transversal through m ; consequently its curvature at m, multiplied by w-2, is precisely the sum of the curvatures sought. Now the diameter of curvature in a conic is to the chord intercepted on the normal, as the rectangle under the segments of a parallel to the tangent is to the rectangle under the segments of the normal chord. Hence, if mp, be any radius vector of the conic, ^ij?2 a chord parallel to the tangent at TO, the radius of curvature is known as soon as the point p.^ has been constructed. To effect this, take any circle tangent to the conic at m; this circle and the conic being homological, their axis of homology may be first found, and then the line homologous ix> pip^; this wUl give the point homologous to ^2) and therefore Pi itself ; in fact, the circle once described, p2 may be found by the ruler alone. We now know the rectangle under the two radii of curvature, and the harmonic mean between them : the radii may therefore themselves be found by a simple and well-known construction. It may be observed that the theory of polar curves leads to a construction for the tangent of a curve line, which is different from M. Chasles', and in fact linear. Through the given point P draw four transversals ; each of these will cut the curve in « — 1 points. Take the harmonic centre of each of these four groups with respect to P, and consider the four points thus obtained as deter- mining a conic section passing through P. Pascal's theorem will then determine the tangent to this conic at P ; that is, the tangent required. It is unnecessary to give the reciprocal construction, which enables us, when a curve of the w* class is given tangentially, to determine with the ruler alone the point of contact on any one of its tangents, supposed not to be a double tangent. It should however be added, that a method for the linear solution of these two problems has been long since given in a different and less explicit form by M. Poncelet, in his excellent memoirs on the Analysis of Transversals. The radius of curvature of any point is, of course, by its nature, incapable of linear construction ; but if we imagine ourselves to have constructed the normal at any point, and to have determined on it the centre of curvature of the given curve or of any one of its superior or inferior polar curves ; and if, in addition, a line parallel to the normal be given, in order that the point at infinity on the normal may be known ; we can find linearly the centre of curvature of every single cui-ve ON SOME GEOMETRICAL CONSTEUCTIONS. 29 of the polar system continued as far upwards as we please. This is a consequence of the following theorem : The distances of the centres of curvature of the successive polar curves from their common tangent form a harmonic progression, commencing at infinity and having the given point for its point of evanescence. All the preceding methods admit of an easy application to the theory of surfaces. For example, to determine the tangent plane at m, we must draw three transversals mp, mq, mr, not in one plane, and then proceed as the case of plane curves. If the surface be of the 'nP^ order, its tangent plane will determine on it a curve of the same order ; the point m being a double point in that curve nodal, cuspidal, or conjugate, according as the contact is hyperbolic, parabolic, or elliptic. Taking the last case, we must determine two conjugate semidiameters of an ellipse concentric, similar, and similarly placed with the evanescent conic in the tangent plane, and therefore with the indicatrix of the point m : the directions and magnitudes of the semiaxes of this ellipse may now be deduced by a construction of extreme simplicity (vide Note xxv. on M. Chasles' History of Geometry), and therefore the ratio of the principal curva- tures, and the traces of the principal normal sections on the tangent plane are known. If now we construct the radius of curvature in either of these normal sections, the square of one semi-axis of the indicatrix is found, and therefore that curve may be regarded as completely determined. It hence appears, that to find the tangent plane and the indicatrix of any point, it is requisite to draw sixteen transversals ; not that so many are absolutely essential, but the trouble is rather increased than lessened by taking fewer. From their connexion with the present subject the following geometrical demonstrations of Meunier's and Euler's theorems on curvature may find a place here. If we take a point P on a curve line, and if we consider an evanescent chord drawn parallel to the tangent at P as an infinitesimal of the first order, this chord wiU be bisected by the normal at P ; that is to say, it wiU be divided into two segments whose difference will be infinitesimal of the second order. Moreover, if we take any sagitta perpendicular to the chord, and intersecting it in a point distant only by an infinitesimal of the second order from its centre, it is readily seen that the square of either segment of the chord, divided by the sagitta, may be taken to represent the diameter of curvature of the evanescent arc. Hence, if we take two plane sections of a surface intersecting in an evanes- cent chord, the radii of curvature of the evanescent arcs are inversely as any two sagittse perpendicular to the chord, and bisecting it approximately. If, now, one of the sections be a normal one, we may take for the sagitta in that section the 30 ON SOME GEOMETRICAL CONSTRUCTIONS. intercept on the normal to the surface. Consequently the triangle fonned by joining the extremities of the two sagittse will be right-angled ; and therefore the radius of the oblique section is equal to the orthogonal projection of the normal radius upon the plane of the oblique section, which is Meunier's theorem. It follows also from what has been said, that if we draw a plane parallel to a tangent plane, and distant from it by an infinitesimal of the second order, the curve surface will, in general, determine upon this plane an evanescent hyperbolic or elliptic oval ; and that the chords of the oval, being themselves infinitesimals of the first order, will be bisected within an infinitesimal of the second order at the point at which the normal meets the plane, and which we will call C. We may therefore consider the oval as a central curve having its centre at C : it only remains to shew that it is a conic section. This may be done as follows : Every transvereal passing through C and lying in the plane of the oval, will cut the surface in two points belonging to the oval, and in n — 2 points whose distance from C is infinitely great compared with that of the two first points. Now, if for a moment we consider the diameters of the oval to be finite, the remaining n — 2 points wiU lie at infinity, and therefore an infinitely magni- fied representation of the section we are considering would consist of a finite central conic, replacing the oval, and of the line at infinity n — 2 times repeated, replacing the w — 2 branches which lie at a finite distance from C. Since, then, the radii of curvature of the normal sections vary as the squares of the diameters of the evanescent oval, they vary as the squares of the central radii vectores of a conic section. K there be a double line upon the surface we can construct the two tangent planes at any point m by taking two plane sections passing through m and constructing the tangents of the double points. Each of these tangent planes will cut the surface in a curve having a triple point at m ; but as the direction of one of the three tangents is known a pricnn, being the intersection of the tangent planes, the directions of the remaining two may be found by the construction used for double points ; consequently the directions of the tangents to the principal sections on each sheet of the surface are known, and the principal radii of curvature may be determined by the construction before given for finding either radius of curvature at a double point. The two indicatxices at the point m may therefore be considered as ascertained in magnitude and position. K the osculatmg plane and radius of curvature of the double line itself be required, they may be obtained very simply by a method to be given below. ON SOME GEOMETRICAL CONSTRUCTIONS. '81 If the singular line be of the r^^ order (r > 2), a little consideration will shew that though we cannot determine the tangent planes by any elementary construction, yet, if we assume these planes as known, the indicatrices upon each sheet may be found as easily as if the point were not singular. This is the more remarkable, since the expressions given by analysis for the principal radii of curvature at such points appear to be of great complexity. Let us now take a point m on a curve surface where two sheets of the surface meet and have a common tangent plane. This tangent plane will intersect the surface in a curve having a quadruple point at m ; but the direc- tions of the four tangents may always be ascertained by a quadratic construction. For at such a point the polar surface of the third order will resolve itself into the tangent plane and into a surface of the second order. And it may be shewn that the two generatrices (real or imaginary) of that surface which lie in the tangent plane are in involution with the two pair of asymptotes of the two indicatrices ; that is, with the four tangents before mentioned. Now these two generatrices may be determined by means of the theory of homological figures, since that theory enables us to assign a pair of semi-conjugate diameters of a section of the surface of the second order parallel to the tangent plane at m, whence the directions of the asymptotes of that section become known, and therefore the two generatrices required. The problem now will be : Given four points in a line PQRS, and the four products P61.P62.Pd3.P6i, &c., determine ^i) ^2) ^3» ^4) a pair of points G^, G^ being also given which form an involution with the two pairs 6^, 6^ and ^3, 0^. It is plain that, A being any point whatever, any symmetrical function of the distances A61, A62, A63, Ad^, may be constructed. Hence H^, H^, the harmonic centres of the four points 6-^,62,63, O4 with respect to Gi, G2, are known. But H^, H2 form a pair of points in involution with the two pair sought • and therefore H^, H2 together with 6r,, G2 completely determine the involution. Therefore the centre and foci of the system are known, and con- sequently 01, 62 and 63, 04 may be now quadratically determined. Points of the nature here considered may exist isolated on a curve surface ; but if there be a continuous series of them, we shall have a line along which two sheets of the surface envelope one another (not a cuspidal line, for any transversal plane will determine a section having not cusps, but points of osculation at its intersections with the singular line), and at any point on such a line the two generatrices before mentioned wUl be found to coincide : and consequently the surface of the second order will degenerate into a cone. The side of this cone, existing in the tangent plane at m, may be determined by proceeding as in the general case : 82 ON SOME QEOMETRICAI, CONSTRUCTIONS. and then, instead of a pair of points in involution with 61,6.^,63, 0^, we shall have one focus of that involution given. The second focus may next be constructed (being the harmonic centre of 61, 6^, 63, 64, with respect to the given focus), and the problem becomes quadratic as before. Lastly, let there be a point on a curve surface having a tangent cone of the second order. It will be possible to determine three conjugate diameters of that cone. For, take any two planes passing through the vertex, and having con- structed the tangent lines of the double points in those planes, take the harmonic conjugates of the line of intersection of the two planes with respect to each pair of tangents. This will give the plane conjugate to the line of intersection ; and by taking any two lines harmonically conjugate with respect to the two tangent lines existing in that plane, we shall obtain the directions of the three semi- diameters required. Likewise, their ratios, or rather the ratios of their squares, may be found, since the two sides of the cone in each conjugate plane may be constructed. Hence, we may deduce the directions and the ratios of the squares of the principal semiaxes of the cone. But this determination, involving the solution of a cubic equation, requires the construction of a conic, and is con- sequently not within the limits of elementary geometry. (Vide the Note on M. Chasles' History, already quoted, and a paper by Mr. Townsend in this JoumaJ.) K a curve of double curvature be given in space the principles of the theory of transversals are not immediately applicable : but if we regard it as the inter- section of two geometrical surfaces completely given, we may immediately find its tangent, oscillating plane, and radius of curvature. The tangent at m is of course determined by the intersection of the two tangent planes ; and if we take the two normal sections containing that tange];it, and, having constructed their radu of curvature, let fall a perpendicular from m on the line joining the two centres, this perpendicular will represent in magnitude and direction the radius of curvature of the given curve. This (it will be seen) follows at once from Meunier's theorem, or from that known as Hachette's. III. DE COMPOSITIONE NUMERORUM PRIMORUM FORMAE 4X+1 EX DUOBUS QUADRATIS. [Crelle's Journal, vol. L. pp. 91, 92; 1855.] s IT ?i + q2 + <?3 + . ■+- q» fractio continua, cujus numeratx>r, qui determinanti ?1. 1, 0, 0, . . -1, Qi, 1, 0, . . 0. -1, 93, 1, . . 0, 0, -1, q*, . . . . 1 0, 0. 0, 0, -1, 9» aequalis est, per hujusmodi fonnulam {qiqzqs ... qn-iqn) exprimatur. Erit ergo [?i q^-'-qi-i q^ = fe 5.-i • • • 92 q^ et [g, ... qn'\ = \.q^.q■i ••■ 5.] • fe + i ••• 9n] + [?ig2 ••• 9<-i] • fe+2 ••• ?«] ; quae aequationes pendent ab ilia forma determinantali, ambae autem L. Eulero debentur. Itaque, si quantitatum q par sumatur numerus, ipsaeque ita serie sym- metrica disponantur, ut binae inter se aequales fiant, elucet, quantitatem 84 DE OOMPOSITIONE NUMERORUM PRIMORUM FORMAE 4X + 1, ETC, [^1 9f- <?. 9i--- 9i Qi] summam fore duorum quadratorum inter se primorum ; fit enim [(/,<?3 ••• (7.<?.' ••• 929i] = [7i92 ••• <Z.? + [9i92 ••• ?<-i? •••• Contra in numero quotientium impari, erit [gr, ... qi-,qiqi-i ... 22?i] = (?i •.• 9.-i) • |[?i ••• 9f] + [<Zi ••• 94-2]], unde colligis, numerum [g', ... gf,- ... 9,] primum esse non posse, nee duplicem numeri primi ; si quidem casus excipis, in quibus, aut i unitati aequatur, aut I binario, q unitati. Sit^ numerus integer datus ; jUj, /«2. ••• M, series numerorum, qui ad p primi sunt, ipsiusque p dinaidio minores. V V IP Formentur fractiones continuae — , — , ...— ; quae omnes ita terminentur, Ml M2 M, ut is quotiens qui in extreme loco ponatur unitatem superet. Hinc patet, quanta ftierit numerorum fii,Hi, ...Hg multitudo, tantum fore numerum determi- nantium [^i ... </„], qui dato numero p aequales erunt, neque praeter illos ullum dare ejusdem formae determinantem, cujus et primus et extremus quotiens unitate major sit, quique numero p aequalis esse possit. Jam vero, quum duo determinantes [g"! . . . q^ et \c[„ ... q^ aequales sint, quumque ipsum q„ unitate majus sit, apparet [g",, ...q^ ex una aliqua fractionum - oriri. Unde sequitur, data quavis fractione -, inveniri posse aliam in eadem serie, quae quotientes eosdem, ordine inverse, repraesentet. Sit p primus, formae 4 \ + 1 ; ut numerus determinantium ipsi p aequalium par exLstat. Quum ipse p unus e determinantium serie fiat, unus certo alius inveniri poterit in quo quotientium ordo invertendo non mutatur. Cum sit ergo p = [q,q^...qiqi...q^q;\ erit denique p = [q^q^... g,]^ + fe 5-2 . . . qi _ 1]^. Quam theorematis Fermatiani demonstrationem maxime elementarem esse patet, quum pendeat a conversione fractionum vulgarium in fractiones continuas. Singulos autem formae 1+x^ divisores ex duobus quadratis conflari, eodem modo demonstrare in promptu est. Sit enim /uv = 1 + a;*, apparet fore ^ = [^i 92 • • • qt qi--- q-z qi] »'=fe?3 — g'<g'<...g'8g2] x = [q,qi...qiqi...q^], OxFOBD, Maio 1854. IV. ON THE HISTORY OF THE RESEARCHES OF MATHEMATICIANS ON THE SUBJECT OF THE SERIES OF PRIME NUMBERS. [Proceedings of the Ashmolean Society, Vol. III. No. xxxv. pp. 128-131. Read March 2, 1857.] It is probable that the Pythagorean school was acquainted with the definition and nature of prime numbers ; nevertheless the arithmetical books of the elements of Euclid contain the oldest extant investigations respecting them ; and, in particular, the celebrated, yet simple, demonstration that the number of the primes is infinite. To Eratosthenes of Alexandria, who is for so many other reasons entitled to a place in the history of the sciences, is attributed the invention of the method by which the primes may successively be determined in order of magnitude. It is termed, after him, ' the sieve of Eratosthenes ; ' and is essentially a method of exclusion, by which all composite numbers are suc- cessively erased from the series of natural numbers, and the primes alone are left remaining. It requires only one kind of arithmetical operation ; that is to say, the formation of the successive multiples of given numbers, or, in other words, addition only. Indeed it may be said to require no arithmetical operation whatever ; for if the natural series of numbers be represented by points set off at equal distances along a line, by using a geometrical compass we can determine without calculation the multiples of any given number. And it was in fact by a mechanical contrivance of this nature, that M. Burckhardt calculated his table of the least divisors of the first three millions of numbers. But simple as this process is, the questions to which it gives rise are among the most obscure of the theory of numbers. F 2 36 ON THE HISTORY OF THE RESEARCHES OF MATHEMATICIANS Adopting (with a slight variation in its meaning) an expression introduced by M. PoUgnac, we may call the series of numbers left unerased, after the erasure of the multiples of any given primes, the diatomic series of those primes. Thus the diatomic of 2.3.5 is 1, 7, 11, 13, 17, 19, 23, 29 : and it is unnecessary to continue the series further, because if a denote any one of the eight numbers we have written down, the remaining terms of the series are included in the eight arithmetical progressions 30m + a. In general, if Q denote the product of any given primes, in forming their diatomic we need only attend to the terms less than Q. A few of the properties of this finite series are very easily seen. In the first place, the number of the diatomic terms is that of the numbers less than Q and prime to it, or O . (P — 1), if P be any one of the given primes and 11 denote a continued product. Secondly, the diatomic terms are distributed sym- metrically ; that is to say, if a be a diatomic term, Q — a is one too. The sum of the diatomic terms is therefore iQri(P— 1) or inP(P— 1). It is not difficult similarly to form expressions giving the sums of any positive and integral powers of the diatomic numbers ; or series giving expressions convergent up to a certain point, for their product or the powers of their reciprocals. It is therefore possible to form an equation of which the coefficients are functions of the given primes, and the roots are the diatomic numbers. But it does not appear that this equation throws much light on the nature of its roots. A remark of greater interest is due to Legendre. Let us denote by (r) the greatest whole number not surpassing a given positive numerical quantity r ; let «i t*2«3 ... be the diatomic terms of any given primes, and let a be a numerical quantity inferior to % + ], but not inferior to u^ ; then The series is to be continued till it stops of itself, and the signs of summation 2 extend to every possible combination of the given primes PiP^ ... taken one by one, two by two, &c. The principle on which the demonstration of this equation (and of many resembling it, which occur in the theory of numbers) is founded, may be termed the principle of cross classification, and may be enunciated thus. If o-j 0-2 • . . be a system of cross-classifying classes, and if (o-j) denote the number of things in the class o-j, (o-j o-^) the number of things common to the two classes 0-, and 0-2, (o-j 0-2 cr^) the niunber of things common to the thre*? classes a-i, a-2, a^ : then 2(<r) — 2((rjor2)-|-2(o-i(T2(r3) — ... will express the whole number of things present in the classes o-j, o-^ ... . Legendre's formula, it is readily seen, assigns the index k of any given i ON THE SUBJECT OF THE SERIES OF PRIME NUMBERS. 87 diatomic term %. Conversely we can express % in terms of k. Let us denote the function v/«^ v/ « \,v/ « \ v. ^/ ^ and let us form the series of terms ^{k) = ai, c}>{k + ai) = a2, (p{k + a2) = a3... till we arrive at last (as we shall certainly do) at two consecutive terms equal to one another, say a„ and a„ + 1 ; we may then stop, for we should find ^« + 1 ~ <^n + 2 ~ ^n + 3 ~ • • • • The expression for ttj. will then be Ut = k + a„, or iik = k + (p{k + (p{k + (j)(k+ ...))). It must be confessed that this result is one which, for diatomics derived from a numerous group of primes, would involve far too much labour to be of any use. But it is of some slight theoretical interest. For if the given primes PiP^ ... P^ be the x first primes in order of magnitude, it is clear that their 2^, 3^, ... diatomic terms will be Pj,^.i Pj,^2 •••> and that the first diatomic term which will not be a prime is P^^ + i- Given therefore the first x primes, the formula prescribes a direct method for the calculation of the primes intermediate between P^ and P^x + v Thus let the given primes be 2, 3, 5, and let it be required to determine the next prime after 5, we find <^(2) = 1, <^(2 + l) = 2, <^(2 + 2) = 3, 0(2 + 3) = 4, ,/,(2 + 4) = 5, «^(2 + 5) = 5. We have therefore «2 = 2 + <^(2 + <^(2 + <^(2 + ...))) = 2 + 5 = 7. It may be added, that the calculation of the functions (p does not absolutely require the knowledge of the primes Pi.-.P^; only the arithmetical operations which would be requisite for determining their value would involve more trouble than the determination of the primes Pj . . . P,.. V. REPORT ON THE THEORY OF NUMBERS. Pakt I. [Report of the British Association for 1859, pp. 228-267.] 1. 1 HE • Disquisitiones Arithmeticae ' of Karl Friedrich Gauss (Lipsiae, 1801 {ed. 1. 1798}*) and the 'Theorie des Nombres' of Adrien Marie Legendre (Paris, 1830, ed. 3) are still the classical works on the Theory of Numbers. Nevertheless, the actual state of this part of mathematical analysis is but imperfectly repre- sented in those celebrated treatises. The arithmetical memoirs of Gauss himself, subsequent to the pubUcation of the ' Disquisitiones Arithmeticae ;' those of Cauchy, Jacobi, Lejeune Dirichlet, Eisenstein, Poinsot, and, among still living mathematicians, of MM. Kummer, Kronecker, and Hermite, have served to simplify as well as to extend the science. From the labours of these and other eminent writers, the Theory of Numbers has acquired a great and increasing claim to the attention of mathematicians. It is equally remarkable for the number and importance of its results, for the precision and rigorousness of its demonstrations, for the variety of its methods, for the intimate relations between truths apparently isolated which it sometimes discloses, and for the numerous applications of which it is susceptible in other parts of analysis. ' The higher arithmetic,' observes Gauss f, confessedly the great master of the science, 'presents us with an inexhaustible store of interesting truths, — of truths, too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly * The additions enclosed in { } are taken from manuscript notes in the author's interleaved copy ; they are all in his own handwriting. + Preface to Eisenstein's ' Mathematische Abhaudlungen,' Berlin, 1847. Art. 3.] EEPORT ON THE THEORY OF NUMBERS. 39 unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easUy discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts ; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.' 2. It is the object of the present report to exhibit an outline of the results of these later investigations, and to trace (so far as is possible) their connexion with one another and with earlier researches. An attempt will also occasionally be made to point out the lacunae which still exist in the arithmetical theories that come before us; and to indicate those regions of inquiry in which there seems most hope of accessions to our present knowledge. In order, however, to render this report intelligible to persons who have not occupied themselves specially with the Theory of Numbers, it will be occasionally necessary to in- troduce a brief and summary indication of principles and results which are to be found in the works of Grauss and Legendre. It is hardly necessary to add that we must confine ourselves to what we may term the great highways of the science ; and that we must wholly pass by many outlying researches of great interest and importance, as we propose rather to exhibit in a clear light the most fundamental and indispensable theories, than to embarrass the treat- ment of a subject, already sufficiently complex, with a multitude of details, which, however important in themselves, are not essential to the comprehension of the whole. 3. There are two principal branches of the higher arithmetic : — the Theory of Congruences, and the Theory of Homogeneous Forms. The first of these theories relates to the solution of indeterminate equations, of the form ^ a„ x" + a„_, af~^ + ...+aiX + aa = Py, in which a„a„_i ... Oittj and P are given Integral numbers, and x and y are numbers which it is required to determine. The second relates to the solution of indeterminate equations of the form F{xiX.i...x^) = M, in which M denotes a given integral number, and F a homogeneous function of any order with integral coefficients. In this general point of view, these two theories are hardly more distinct from one another than are In algebra the two theories to which they respectively correspond, — the Theory of Equa- tions, and that of Homogeneous Functions ; and It might, at first sight, appear as if there was not sufficient foundation for the distinction. But, In the present 40 REPORT ON THE THEORY OF NUMBERS. [Art. 4. state of our knowledge, the methods applicable to, and the researches suggested by these two problems, are sufficiently distinct to justify their separation from one another. We shall therefore classify the researches we have to consider here under these two heads ; those miscellaneous investigations, which do not properly come under either of them, we shall place in a third division by them- selves. (A) Theory of Congruences. 4, Definition of a Congruence. — K the difference between A and B be divisible by a number P, ^ is said to be congruous to B for the modulus P ; so that, in particular, if ^ be divisible by P, A is congi-uous to zero for the modulus P. The symbolic expressions of these congruences are respectively A=B, mod P, A=0, mod P. Thus 7 = 2, mod 5 ; 13 = - 3, mod 8. It will be seen that the definition of a congruence involves only one of the most elementary arithmetical conceptions, — that of the divisibility of one number by another. But it expresses that conception in a form so suggestive of analogies with other parts of analysis, so easily available in calculation, and so fertUe in new results, that its introduction into arithmetic (by Gauss) has proved a most important contribution to the progress of the science. It wUl be at once evident, from the definition, that congruences possess many of the properties of equations. Thus, congruences in which the modulus is the same may be added to one another ; a congruence may be multiplied by any number ; each side of it may be raised to any power whatever, and even may be divided by any number prime to the modulus. 5. Solution of a Congruence. — If ^ (x) denote a rational and integral func- tion of X with integral coefficients (we shall, throughout this report, attach this meaning to the functional symbols F, f <p, &c., except when the contrary is expressly stated) ; the congruence (p (x) = 0, mod P, is said to be solved, when all the integral values of x are assigned which make the left-hand number of the congruence divisible by P ; i.e. which satisfy the indeterminate equation 4> (x) = Py. It is evident that if a; = a be a solution of the congruence ^ {x) = 0, every number included in the formula x = a + tJLP is also a solution of the con- gruence. But the solutions included in that formula are aU congruous to one another and to a. It is proper, therefore, to consider all these congruous solu- tions as identical, and in speaking of the number of solutions of a congruence d Art. 7.] REPORT ON THE THEORY OF NUMBERS. 41 to understand the number of sets of incongruous solutions of which it is sus- ceptible. To assign, by a direct method, all the solutions of which a proposed congruence is capable, is the general problem which, in the Theory of Numbers, corresponds to the problem of the solution of numerical equations in ordinary algebra. But the solution of the arithmetical problem is attended with even greater difficulties than that of the algebraical one ; and the attention of geo- meters has been turned with more success to the improvement of the indirect or tentative methods of solution, and to the discovery of criteria of possibility or impossibility for congruential formulae, than to their direct solution. It is to be observed that, by virtue of a remark already made, the tentative solution of a congruence involves no theoretical difficulty. For i£ x = a be a solution, every number included in the formula x = a + ixP is also a solution, and among these numbers there is always one, and only one, comprised within the limits and P-1 inclusively. By substituting, therefore, for x all numbers in suc- cession less than the modulus, and rejecting those which do not satisfy the congruence, we shall obtain its complete solution. But the interminable labour attending this operation, notwithstanding all the abbreviations in it suggested by the Calculus of Finite Differences, renders its application impossible, except when the modulus is a low number. 6. Systems of Residues. — The set of numbers 0, 1, 2 ... P- 1 (or any set of P numbers respectively congruous for the modulus P to those numbers) is termed a complete system of residues for the modidus P. By a system of residues prime to P, we are to understand a complete system, from which every residue has been omitted which has any common divisor with P. Thus 1, 5, 7, 11, or 1, 5, —5, — 1, are the terms of a system of residues prime to 12. The word Residue is employed instead of Remainder, because the word Remainder would suggest the idea of a positive number less than the modulus or divisor ; whereas it is frequently convenient to consider residues differing from those positive remainders by any multiples of the modulus whatever. 7. Linear Congruences. — The general form of a linear congruence is ax + b =0, mod P ; a, b, and P denoting given numbers, and x a number to be determined. The theory of these congruences may be considered to be complete, both as regards the determination of the solutions or roots themselves and of their number. If a be prime to the modulus, there is always one solution, and one only; if a have a common divisor with the modulus which does not aJso divide b, the congruence is irresoluble ; if ^ be the greatest common divisor of a Q 48 REPORT ON THE THEORY OP NUMBERS, [Art. 8. and P, and if i also divide 6, the congruence has 5 solutions. In every case when the congruence is resoluble, the direct determination of its roots may be made to depend on the solution of a congruence of the form ax = l, mod P, in which a is prime to P. This congruence coincides with the indeterminate equation ox = 1 + Py, methods for the solution of which were known to the ancient Indian geometers*, and have been given in Europe by Bachet de Meziriact, EulerJ, and Lagrange f The methods of these writers ultimately depend on the conversion of a vulgar fraction into a continued fraction, and in one form or another have passed into every book on algebra. Nor would it have been proper to allude to them here, were it not that they serve to supply us with a clear conception of what we have a right to expect in the solution of an arithmetical problem. In such problems, we cannot expect to express the quaesita as (discontinuous) analytical functions of the data. Such expressions may indeed, in many cases, be obtained (by the use of the roots of unity or by other methods) ; but the results of the kind which have hitherto been given, though sometimes of use in calculation, may be said, with few exceptions, to conceal rather than to express the real connexion between the numbers required and the numbers given. The arithmetical solution of a problem should consist in prescribing a finite number of purely arithmetical operations (exempt from all tentative processes), by which all the numbers satisfying the conditions of the problem, and those only, are obtained. It is clear that this description exactly applies to the methods on which the solution of linear congruences depends ; but, unfortunately, the higher arithmetic pre- sents but few examples of solutions of equal perfection. 8. Besides the older methods for the solution of the equation ax=l+Pi/, others have, in very recent times, been suggested. Of these the following may serve as examples : — A. In the equation ax = l+Py, or the congruence aa; = 1, mod P, form * See the Arithmetic of Bhascara, cap. xii, and the Algebra of Brahmegupta, cap. i, in Mr. Colebrooke's translation, London, 1817. + Problimes plaisans et d^lectables, qui se font par les nombres. Seconde Edition. Par Claude Oaspar Bachet, Sieur de Meziriac, Lyon, 1624. (See Props, xv to xxv.) X Comment. Acad. Petropol. torn. vii. p. 46, or in the Collection of Euler's Arithmetical Memoirs (L. Euleri Commentationes Arithmeticae CoUectae, Petropoli, 1849), vol. i. p. 2; and in his Elements of Algebra, part ii. cap. 1. § Sur la Resolution des Problfemes Ind^termin^s du seconde degrd. Hist, de 1' Acad, de Berlin, 1767, p. 165. (See Arts. 7, 8, and 29 of the Memoir.) Also in the Additions to Euler's Algebra, sects, i and iii. (Lyon, an. in.) i Art. 9 ] REPOKT ON THE THEORY OF mJMBERS. 4^ the residues of the successive powers of a for the modulus P. If a be prime to P, we shall at last arrive at a power which has +1 for its remainder or residue. The residue of the power immediately inferior to this power is the value of cc in the congruence ax = l, mod P. This solution is evidently an application of Fermat's Theorem *. B. Let there be P points Ai, A^, ... A^, arranged at equal distances on the circumference of a circle. Join A-^ to A^^i, ^a + i to ^jo + i--- ^i^d. so on con- tinually. It can be proved that if a be prime to P, we shall not return again to Ai, untU we have passed through every one of the P points, and have formed a polygon of P sides. Let X^, X^, ... X^he the vertices of this polygon, taken in order, and let A^ = X„ ^ j ; then x = m is the value of x in the congruence ax = 1, mod Pf. C Let an origin and a pair of axes be assumed in a plane, and let all the points be constructed whose coordinates are integral multiples of the linear unit ; call these points unit points. Join the origin to the point (a, P). If a be prime to P, no unit point can lie on the joining line, but on each side of the joining line there will be a point lying nearer to it than any other. Let (^i >?i), (^2 "72) be the coordinates of these points, and let ^i'-V\<^.i'-'ii', then fj, m, and ^2) '?2 are the least positive numbers satisfying the equations «'/i-i'?i = l, a.72-P^2=-l- The late M. Crelle, of Berlin, in the 45th volume of his Journal (p. 299), has given a very useful table, containing the least positive numbers x^ and x.^ which satisfy the equation a^ x^ — aj ^^2 = 1, for all values of Oi up to 120, and for all values of ag prime to a^ and less than it. 9. Systems of Linear Congruerices. — The theory of these systems is left imperfect in the work of Gauss (see 'Disq. Arith.' art. 37); but, by the aid of a few subsidiary propositions relating to determinants, we may, in every case, obtain directly all possible solutions of any proposed system ; and (what is frequently of more importance) we can decide a priori whether a given system of linear congruences be resoluble or not, and if it be resoluble we can assign the f * Biset, BUT la Kdsolution des Equations du premier degr6 en Nombres entiers. (Journal de I'Ecole Polytechnique, cahier xx. p. 289.) Libri, MSmoires de Mathematique et Physique (Florence, 1829), pp. 65-67. Poinaot, Reflexions sur lea Principea Fondamentals de la Th6orie des Nombrea (Paria, 1845), cap. iii. no8. 19 and 20. For another solution by M. Binet, aee Comptes Rendua, xiii. p. 349. See also Canchy, Comptes Rendua, xii. p. 813. {Exer. d'Anal. et de Phys. Math., vol. ii. p. 1.} + Poinsot, Reflexions, &c., cap. iii. noa. 17 and 18, O 2 44 REPORT ON THE THEORY OF NUMBERS. [Art. 9. number of its solutions. The following theorems by which the determination of the number of solutions is, in every case, effected, wUl sufficiently indicate the nature of these investigations. Let the proposed system of congruences be represented by (1, 1) Xi + (1, 2) x^ + {l, 3) a^+ ... + (1, n)x^=Ui, (2, 1) x, + (2, 2) X2 + (2, 3) Xs+ ... +(2, n)x„=Ui, (A) (n, 1) Xi + (n, 2) 0C2 + {n, 3) CC3+ ... +(n, n) x„=u„; let the modulus be q, and the determinant 2 + (1, 1) (2, 2)...(n, n) = D. If the determinant be prime to the modulus, these congruences will always admit of one, and only one, system of solutions, namely, that supplied by the system of congruences A;=n ^^ Dx. = 2 -J— — - Ml. But if D be not prime to q, let q=p^i.p{*i ... where pi, jpi, &c. denote different primes. In order that the proposed system should be resoluble for the modulus q, it must be separately resoluble for each of the modules p^^i , Pi*^ , &c. ; and, conversely, if it be resoluble for each of those modules, and admit P, solutions when taken with respect to the modulus pi*^ , P^ solutions when taken with respect to the modulus p.^^, and so on, it will be also resoluble for the modulus q, and will admit P^xPiXP^.,. solutions for that modulus. It is, therefore, only necessary to assign the number of solutions of the congruences (A), for a modulus p^ which is the power of a prime. Let 7„ be the index of the highest power of p which divides D ; and similarly, let 7, denote the index of the highest power of ^ which divides all the minors of D which are of order r ; then if 7„ — /„_j S m, the system (A) (if resoluble at all) admits of p^" solutions ; but if 7„>m + 7„_i, it will always be possible, in the series of differences ■*n -'n — 1> -'n — 1 "" -'n — 2> •••> to assign a pair of consecutive terms I^^^ — I^, I^ — I^_i, satisfying the in- equaUties 7,,,-7, > m ^ 7,-7,.,; and then the number of solutions (supposing always that the congruences are resoluble) is expressed by the formula p'>- +'"-''''". The analogy of this theory with the corresponding algebraic theory of systems of linear equations is in particular cases very striking. For example, we have in Algebra the theorem : Art. I O.J REPORT ON THE THEORY OF NUMBERS. 45 ' The system of « linear equations {2,l)x, + {2,2)x, + {2,3)xi+...+(2,n)x, = 0, (n, 1) Xi + {n, 2) cc2 + (w, 3) X3+ ...+(n,n) x„ = 0, implies either that Z) = 2 + (1, 1) (2, 2) . . , (w, w) = 0, or else that Xi, X2, ...x„ are separately equal to zero.' In the Theory of Numbers we have the corresponding theorem : ' If n linear and homogeneous functions of an equal number of indetermi- nates be congruous to zero for a prime modulus, either the determinant of the system is congruous to zero for that modulus, or else every one of the indetermi- nates is separately congruous to zero.' 10. Fermat's Theorem. — The theory of congruences of the higher orders is so essentially connected with Format's Theorem, that it will be proper before proceeding further to introduce a few considerations relating to that celebrated proposition. It may be considered from two different (though closely connected) points of view, each of which has proved equally fertile in consequences. First, it may be regarded as asserting that, if ^ be a prime number, and x any number prime to p, the remainder left by the power x^~'^ when divided by p is unity. It is thus the fundamental proposition in the arithmetical theory of the residues of powers, or, which is the same thing, of binomial congruences. Or, secondly, it may be regarded as asserting that the congruence x^~''- = l, mod p, has precisely p — 1 roots ; and that these roots are the terms of a system of residues prime to p. It is in this latter point of view that the theorem is the basis of the general theory of congruences. We may observe that the demonstrations of Fermat's Theorem point to this twofold aspect. The proof, which is found in most English treatises of Algebra (it is the first of those given by Euler*), and which depends on the property of the binomial or multinomial coefficient, would naturally lead us to regard the Theorem in the first point of view. The same may be said of Euler's second * Comment. Acad. Petropol., vol. viii. p. 141, or Comment, Arith., vol. i. p. 21. This is the first demonstration of the Theorem discovered, since the time of Fermat. The memoir containing it was presented to the Academy of St. Petersburg, Aug. 2, 1736. ^0 KEPOET ON THE THEORY OF NUMBEKS. [Art 10, demonstration*, which consists in showing that the index of the lowest power of X in the series 1, x, x\ x\ &c., which leaves unity for its remainder when divided by v> is either p-1, or some submultiple of p-1 ; or agam of the demonstration of MM. Dirichlett. Binett, and Poinsotf, which depends on the observation that the terms of a system of residues prime to any modulus, being multiplied by any residue prime to the modulus, still form a system of residues prime to the modulus. But a remarkable proof of the theorem, in the second expression we have given to it, occurs in a memoir of Lagrange ||. As this proof (though very elementary) has not been copied by subsequent writers, and is consequently but little known, its nature may be indicated here. Let the product (a; + l) (x + 2) (x + S) ... (x + p-1) be represented by X denoting an absolutely indeterminate quantity. Writing x + l for x, and multiplying by a; + l, we obtain the identity (« + l)'' + ^(x + l)''-^ + ^2(^ + ir-'+-+^-i(«' + l) whence, by equating the coefficients of like powers of x, we find _ p(i?-l) ^1- 1.2 ' ^, p(p-l)(i>-2) , (p-l)(f>-2) ^^ ^■^""^ iTO 1.2 " 3^= 1.2.3.4 1.2.3 ' 1.2 (2J-l)4p_l=l+^l + ^2 + -43+ ... +A-2- I * Novi Commentarii Petropol., vol. vii. p. 49, or Comment. Arith., vol i. p. 260. From the point of view in which Fermat presents his theorem, it is not improbable that the demonstration he had found of it was no other than this of Euler's. (See Fermati Opera Mathematica, Tolosae, 1679, p. 163.) It has been adopted by Gauss in the Disquisitiones, Art. 49. t Crelle's Journal, vol. iii. p. 390. ♦ Journal do I'^Icole Polytechnique, Cahier xx. p. 289. § Reflexions sur la Th6orie des Nombres, p. 32. But the principle of this demonstration is employed by Gauss in a memoir published in the Comm. Soc. Gotting. vol. xvi. p. 69, to which we shall have again to refer. (See Art. 19 of this Report.) 11 Demonstration d'un Thtoreme nouveau concemant les Nombres Premiers (Nouveaux Mdmoires de I'Acadimie Eoyale de BerUu, 1771, p. 125). The 'new theorem' is that known as Sir. J. Wilson's. I Art. 11.] KEPOBT ON THE THEORY OF NUMBERS. 4T From these equations we successively infer the congruences Ai = 0, ^2 = 0, Aa = 0, ... Ap_.2=0, and lastly, Ap_i = —1, mod p. We have, therefore, the indeterminate congruence (x + l){x + 2){x + 3) ... (x+p-l) = xP-'^-l, modp, which is evidently identical, i. e. it subsists for all values of x. And since, if ttj, aa,'... ap_i be the terms of any system of residues prime to p, the factors x — Oi, x — a^, x — a^, ... x — ap_i are one by one congruous to the factors x + 1, x + 2, x + 3, ... x+p — 1 taken in a certain order, the products {x — fti) (x — ttz) ... (x — ap_i) and (x + 1) {x + 2) ... (x +p — 1) are also identically congruous for the modulus p, so that we may write (x — Oi) (x — a^) ... {x — ap_i) = «^~' — 1, mod^. This congruence exhibits in the clearest manner possible what the real nature of the function x''~^ — l is when considered with respect to the modulus p, and explains to us why it assumes a value divisible by p, when we assign to x any integral value not divisible by p. It will be observed that the last of the p — 1 congruences included in the congruence (x-l) {x-2){x-3) ...(x-p-l) = xP-'^-l, modp, (which is a particular case of that last written), namely, the congruence 1.2.3 ...J) — ! = —1, mod p, is the symbolic expression of Sir J. Wilson's Theorem. 11. Lagrange's Limit of the Numbei' of Roots of a Congruence. — The full development of the consequences of Fermat's Theorem requires the aid of the following proposition, which was first given, in a slightly different form, by Lagrange *. ' If F (x) be a function of a; of w dimensions, such that F (a) = 0, mod p, then a function of a; of n — 1 dimensions, Fi (x), can always be assigned such that we shall have the identical congruence F(x) = (x — a)Fi(x), mod p.' Hence we may infer that no congruence, of which the modulus is prime, can have more incongruous roots than it has dimensions ; and, if a congru- ence have congruous roots, we obtain a definition of their multiplicity; viz., if F(x) = (x — ay Fi(x), modp, then we may say that F(x) = 0, modp, has * Nouvelle M6thode pour resoudre les Problemes Ind^termin^s en Nombres entiers. (See Hist. Ac. Berl. 1768, p. 192.) The case of binomial congruences of the form a;" = 1 had already been treated by Euler. (See Nov. Comment. Petropol. vol. xviii. p. 85, or Comment. Arith. vol. i. p. 516, Art. 28 of the Memoir.) 48 BEPORT ON THE THEORY OF NUMBERS. [Art. 12. r roots congruous to a. We may also observe that this theorem enables us at once to infer Lagrange's indeterminate congruence from the first expression of Format's Theorem. For since x"-^-! is = for the values x = l, x = 2, ...x = p-l, we may, by successive applications of the preceding theorem, show that x"-^-! = (x-l)(x-2)...(x-p + l),modp. 12. Theory of the Residues of Powers. — The principal elementary theorems relating to the Residues of Powers are the following. They are all due to Euler *, who was the first to demonstrate Fermat's Theorem, and to develope the numerous arithmetical truths connected with it. I. If e and / be conjugate divisors of ^) — 1 so that p — 1 =e/; the con- gruence x^=l, modjp, always admits of / incongruous roots. Let these roots be denoted by Oj, a^, ... af. Then each of the/ congruences x'= a,, admits of e solutions, and the ef roots of these / congruences exhaust completely the p — 1 residues prime to p. It appears, therefore, that if we raise the residues of ^ to the power e, they will divide themselves into / groups of e numbers apiece ; the e numbers of each group giving, when raised to the power e, the same residue for the modulus p. The numbers Oj, ag, ... ay are termed the quadratic, cubic, biquadratic, quintic, &c. residues oi p, according as e = 2, e = 3, e = 4, e = 5, &c., because they are each of them congruous to an e^^ power (and indeed to an e*l» power of e different numbers), and because no other number beside them can be congruous to such a power. Thus every uneven prime has \{p — X) quadratic, and as many non-quadratic residues; every prime of the form 4w-t-l has \{p — l) biquadratic residues, and three times as many non-biquadratic residues, &c. II. It is readily seen that if the same number x satisfy the two congruences a/t = 1, and x-^»=l, it also satisfies the congruence £c^ = l, modp; where d is the greatest common divisor of/i and/. If therefore /be the lowest index for which the number x satisfies the congruence x^ =1, modjj9, / is a divisor * Euler's memoirs on this Theory are : — (i.) Theorematum quorundam ad numeros primes spectantium demonstratio. Comment. Arith. Tol. i. p. 21. (ii.) Theoremata circa residua ex divisione potestatum relicta. Ibid. p. 260. (iii.) Theoremata arithmetica novo methodo demonstrata. Ibid. p. 274. (iv.) Disquisitio accuratior circa residua ex divisione quadratorum aliarumque potestatum per numeros primos relicta. Ibid. p. 487. (v.) Demonstrationes circa residua ex divisione potestatum per numeroa primos resultantia. Ibid. p. 516. i A Art. 13] REPORT ON THE THEORY OF NUMBERS. 49 of p — 1 ; as indeed appears directly from Euler's second demonstration of Fermat's Theorem. Let -^ {/) denote the number of numbers less than f and prime to it; then there are always ^{f) roots of the congruence x^=l, mod p, which cannot satisfy any other congruence of lower index and similar form. These are called primitive roots of the congruence x-^=l, mod^; they are also said to appertain to the exponent/. It f=p — l, the y^{p — l) primitive roots of the congruence x^~^ = 1, vaod p, are termed for brevity (though the de- signation is somewhat improper) the primitive roots of p. There are therefore y^[p — l) primitive roots of ^. 13. Primitive Roots. — The problem of the direct determination of the primitive roots of a prime number is one of the ' cruces ' of the Theory of Numbers. Euler, who first observed the peculiarity of these numbers, has yet left us no rigorous proof of their existence * ; though, assuming their existence, he succeeded in accurately determining their number. The defect in his demonstration was first supplied by Gauss t, who has also proposed an indirect method for finding a primitive root. This method | consists in taking any residue a of p, and determining (by the successive formation of its powers) the exponent / to which it appertains. Jf f=p — l, a is itself a primitive root of p; if not, let 6 be a second residue of p, not contained in the period of a, {i.e. not congruous for the modulus p to any one of the numbers a", a, a^, ... a/~^,) and let the exponent to which h appertains be determined. This exponent cannot (as is shown by Gauss) be identical with, nor yet a divisor of, the exponent to which a appertains ; but it is always possible by a comparison of the values of a and b to determine a third number, c, which shall appertain to an exponent divisible by each of the exponents to which a and b appertain. By proceeding in this way we shall evidently obtain num- bers appertaining to exponents continually higher, till at last we come to a number appertaining to the exponent p — 1 ; i.e. to a primitive root of p. M. Poinsotf proposes the following method. If 2, q^, q^, ... &c. be all the prime divisors of p — 1, raise the numbers ±1, ±2, ±3,... ±i{p-l), which form a system of residues prime to p, to the powers of which the * See the memoir (i) of the preceding note ; and Gauss's criticism on it ; Disq. Arith. Art. 56. t Disq. Arith. Art. 52-55. J Ibid. Art. 73-74. § Reflexions sur la Thdorie dee Nombres, cap. iv. art. 3. H h so REPORT ON THE THEORY OF NUMBERS. [Art. l3- indices are 2, q^, q^, &c. ; so as to determine all the quadratic residues of p, and its residues of the powers qi, q^, &c. If from the system of residues 1,2, 3, ... p-1, we successively exclude these residues of squares and higher powers, we shall have \|'(jp— 1) numbers left, which cannot be congruous to any power having an index that divides p — 1, and which are consequently (as may easUy be shown) the primitive roots of p. This method is very symmetrical ; and if the problem proposed be to find all the primitive roots of p, it is sufficiently direct. But it is (like many other direct methods in the Theory of Numbers) of interminable prolixity; and becomes absolutely impracticable if ^ be a number even of moderate size, as it requires us to form the residues of the successive powers of the numbers 1, 2, 3, ... ^(j9 — 1). Of course, in performing this operation, the multiples of jo are to be rejected as fast as they arise ; but, notwithstanding this abbreviation, and others which a little experience will readily suggest. Gauss's method is, for any practical purpose, greatly preferable. In a memoir by M. Oltramare in Crelle's Journal (vol. xlix. p. 161), several considerations are offered for facilitating the determination of the primitive roots of primes in numerous special cases. Some, however, of the general results of this memoir are erroneous, at least in expression, and the demonstrations of the more particular conclusions contained in it involve no new principle, but may be obtained by combining the definition of primitive roots with the criteria by which (as we shall hereafter see) we are enabled to decide on the quadratic or cubic characters of the residues of given primes. The following may serve as examples of the very interestmg results which are thus obtained by M. Oltramare : — 'K a be a prime number and 2a + 1 be also a prime, 2 or a is a primitive root of 2a + l, according as a is of the form 4n + l or 4?n-3.' Thus 2 is a primitive root of 83, 11 is a primitive root of 23, 83 of 167, &c. ' If a be a prime number, other than 3, and if ^ = 2'"a + 1, where m is > 1, be also a prime, 3 is a primitive root of p, unless the conginience 32™-! + 1=0, mod j», be satisfied.' Thus 3 is a primitive root of 89, and of 137. Theorems of the same character will be found in the ' Thdorie des Nombres * ' of M. Desmarest. By their aid M. Desmarest has constructed a table giving a primitive root for every prime less than 10,000. * Paris, 1852. See pp. 275-279. Art. 14.] REPORT ON THE THEORY OF NUMBERS. 51 14. Indices. — If 7 be a primitive root of p, the least positive residues of the p — 1 successive powers of 7, y^ ~2^ y^^ ... ■yi'-^^ 7*"*, which we may denote by 71, 72, 73, ... 7j,_2, 1, are all incongruous for the modulus p. These residues, therefore, irrespective of the order in which they occur, coincide with the numbers 1, 2, 3, ...p — 1, i.e. they represent the terms of a complete system of residues prime to p. If 7* = a, mod p, then k, or any number congruous to k for the modulus p - 1, is termed the index* of a for the primitive root or base 7; and this is expressed symbolically by writing (c = Inda, mod(|) — 1), or k = Ind^^a, mod(p — 1). The principal properties of these indices, which it is clear are a kind of arithmetical logarithm, are as follows : — (1) Ind {AB) = Ind ^ + Ind B, mod (p - 1). (2) Ind {A") = <T Ind A, mod {p - 1). (3) Ind (^ , mod _p) = Ind ^ - Ind B, mod {p-\). /A \ [The symbol ( -^ , mod jA is used to denote the value of x deduced from the congruence Bx = A mod p.] (4) Ind^ A = Ind.^ 7'. Ind^, A, mod (p - 1). (5) I(A = B, modp, Ind A = Ind B, modp - 1. In these congruences A and B represent numbers prime to p, <r any integral number, and 7 and 7' two different primitive roots. The great importance of these indices in arithmetical researches has induced the Academy of Berlin to publish a volume containing tables of the numbers corresponding to given indices, and of the indices corresponding to given numbers for all primes less than 1000. This volume, the ' Canon Arithmeticust,' was edited by C. G. J. Jacobi, and contains, besides the Tables, a preface * The reader must be careful to distinguish between the index of a number and the exponent to which the number appertains. The exponent does not depend on the choice of the primitive root : for » — 1 a given number it has but one value, a, which is such that is the greatest common divisor of the index and of p—l. The index may have any one of ^ (a) different values; which of these it has depends on the particular primitive root chosen, t Berlin, 1839. H 2 62 REPORT ON THE THEORY OF NUMBERS. [Art. 14. explaining the methods which he adopted in their construction. The annexed specimen will serve to exemplify the arrangement of the Tables : — p = 29, _p-l = 2«.7. Numeri. Indices. L 10 1 2 3 4 5 6 7 8 9 N. 1 2 3 4 5 6 7 8 9 1 10 13 14 24 8 22 17 25 18 28 11 27 22 11 10 20 5 26 1 6 2 20 26 28 19 16 15 6 21 1 1 23 21 2 3 17 16 7 9 15 2 7 12 4 11 23 27 9 3 1 ■ 2 12 19 6 24 4 8 13 25 14 M. Burckhardt, to whom arithmetic is indebted for an excellent Table of the divisors of numbers from 1 to 3,036,000*, has inserted in his work, and apparently only to fiU up a blank-page at the end of the first million, a table stating the number of figures in the decimal period of the fraction -, for every prime number p less than 2500. It is evident that the number of terms in the decimal period of - is nothing else than the exponent to which 10 appertains for the modulus p. M. Burckhardt's Table, therefore, at once apprises us that out of the 365 primes inferior to 2500 (2 and 5 are not counted in this enume- ration, as being divisors of 10), 10 is a primitive root of 148 ; because there are 148 primes p below 2500, the reciprocals of which have decimal periods con- sisting of^ — 1 figures. Again, for 108 of the remaining primes below 2500, the exponent to which 10 appertains is ^{p — 1). Of these 108 primes, 73 are of the form 4 w -f- 3, from which it may be inferred that - 1 is a primitive root of those 73 numbers. M. Burckhardt's Table supplies us, therefore, with a primitive root (and that root the most convenient for the purposes of compu- tation) of 148 + 73 = 221 out of the 365 primes inferior to 2500. Nor is this the limit to its usefiilness ; for when the exponent to which 10 appertains is as high a8^(^ — 1) or ^{p — 1) or i(p — 1), it is possible by methods which Jacobi has indicated to construct the Table of Indices with very little labour. Jacobi says that had it not been for this table of Burckhardt's he should hardly have ventured on the construction of the 'Canon Arithmeticus,' on * Paris, 1814-1817. A Table containing the exponents to which 10 appei-tains, for every prime less than 10,000, has since been given by M. Desmarest (See p. 308 of his ' Th^orie des Nombres.') Art. 14.] REPORT ON THE THEORY OF NUMBERS. 53 account of the prolixity and uncertainty of the tentative methods for the in- vestigation of primitive roots. But, while endeavouring to avail himself of the results of M. Burckhardt's Table, for the computation of his own Tables of Indices, in other cases besides those in which that Table immediately fur- nishes a primitive root, he was led to the invention of a general method of procedure, which, as he says, would have enabled him to dispense with the assistance of Burckhardt's Table altogether, or to extend his Canon to any higher limit which the expense of printing would have admitted. This method is not in principle very different from Gauss's process for finding primitive roots, but the form which Jacobi has given to it possesses great advantages, for the purpose to which he has applied it. He first of all takes a number a (not quite at hap-hazard, for quadratic residues can at any rate be excluded by the law of reciprocity; see inf. Art. 16); and determines its period of residues, and the exponent a to which it appertains. Let aa = p — 1, and let the residues of a, a^, a?, ... a" be entered in a Table of which the arguments are the indices 1, 2, 3, ...j9 — 1, opposite to the indices, a, 2a , 3 a', ... aa, respectively. It has been shown by Gauss that there are always ^ , . primitive roots for which this assignment is true. A number h is then taken, not contained in the period of a, and the residues of its successive powers are formed tiU we come to the lowest power of it that is congruous to any power of a ; so that b^ = a^, mod p. Let /3 be the exponent to which h appertains, 6 the greatest common divisor of a and /8, and X = -^ their least common multiple ; let also /3/3' = p — 1. It may a h be proved that B= ; A=— ; where k is some number less than 6 and prime 6 6 to it, so that ^ is the greatest common divisor of A and a. These relations show, that when we know the numbers a, A, and B, we can immediately find 6, k, and /8, without having to raise h to any power higher than h^. We may then assign to h any index of the form l(i', where I is prime to /3, and congruous to k for the modulus Q. The number of such values of I (incongruous for the modulus (8) is , , J. ; and, whichever of them we take, there will be j-t-t — y(0) yW primitive roots, for which h will have the index l^, while a retains the index a'. We must next form the residues of the X — a products included in the formula a^ 6"; where x has any value from 1 to a inclusive, and y any value from 1 to B—1. These residues are all incongruous ; the indices of aU of them are known ; 64 REPORT ON THE THEORY OF NUMBERS. [Art. 14. and, together with the a powers of a already entered in the table, they exhaust p — 1 all the numbers which have indices divisible by ^—r — In practice, it will almost always happen that X is equal to jp — 1. When this is so, nothing remains to complete the operation but to enter in the Table the residues of the numbers a' 6* opposite to the indices corresponding to them. p-1 But, if \ < p — 1, we may take that residue which has -^-r — for its index, and use it to replace a in the preceding operation, while b is replaced by some other residue not yet entered in the Table. In this way we shall ultimately (and in practice very speedily) obtain a complete Table of Residues corresponding to 'given indices, which, of course, immediately supplies us with the inverse Table of Indices corresponding to given residues. It will be seen (as has been already observed) that the process is not dissimilar to Gauss's method for determining a number appertaining to the exponent X when we already know two numbers a and b appertaining to the exponents a and /3 respectively. But it is so arranged by Jacobi that hardly a single figure is wasted, the primitive root, instead of being found by a preliminary investigation, presenting itself at the end of the operation, and being recognized by its standing opposite to the index 1. To calculate with rapidity the residues of the powers of a number, Jacobi employs a method proposed by M. Crelle in his Journal, vol. ix. p. 30, and which is most easily explained by an example. Let p — 11, and let it be required to determine the residues of the powers of 3 ; and the residues of those powers multiplied by 7. Column I. 1,2, 3, 4, 5, 6, 7, 8, 9, 10 ; „ II. 3,6,9.1,4,7,10,2,5, 8; „ III. 3,9,5,4,1; „ IV. 10, 8, 2, 6, 7. The first column contains the numbers 1, 2, 3, ... ^ — 1. The second column begins with 3 (the number the powers of which we are considering), and consists of numbers formed by successive additions of 3, multiples of 11 being rejected as fast as they arise. The third column also commences with 3, and is so formed that any number r in it is followed by the number which in column II. stands under r in column I. This column contains the residues of the powers of 3 taken in order, and stops at 3* because after that the same residues recur. Art. 16.] EEPORT ON THE THEORY OF NUMBERS. 55 Lastly, column IV. begins with 10 (the number which in column II. stands under 7 in column I.), and is formed in the same way as column III. It represents the residues of 7.3, T.S^, &c 15. Quadratic Residues. — It appears from the theorems cited in Art. 12, that the numbers 1, 2, 3, ...^ — 1 di^dde themselves into two classes of Qua- dratic Residues and Quadratic non- Residues, comprising ^(p — 1) numbers each. Every quadratic residue a satisfies the congruence 05^'*"^' = 1, mod^ ; every qua- dratic non-residue h satisfies, instead, the congruence a:^'^~^' = — 1, mod p. Again, for every quadratic residue the congruence x^ = a, mod p, is resoluble ; for every non-quadratic residue the congruence x^ = b, mod j^, is irresoluble. The solution of almost every problem relating to the indeterminate analysis of quadratic - functions involves a congruence of the simple form x^ = A, mod p. It is there- fore of great importance to obtain a criterion which shall enable us to determine a priori whether a given number is or is not a quadratic residue of a given prime. If we have a Table of Indices for the given prime, we have only to see whether the index of the given number is even or uneven ; if even, it is a quadratic residue ; if uneven, it is a quadratic non-residue. Or, again, we may raise the given number a (by M. Crelle's method, or any other) to the power ^(j9 — 1), and see whether the residue is -1-1 or —1. It is usual to denote the positive or negative unit which is the remainder of <x2*p~'', modp, by the symbol (-), which is known as ' Legendre's Symbol ' ; so that in every case ai(i>-i) = ^-^, jaodp, and (-)= +1 or = -1, according as a is or is not a quadratic residue of p. It will be seen that we also have in every case the equation If a instead of being prime to p be divisible by p, it is convenient to attribute to (- j the value zero. 16. Legendre's Law of Reciprocity. — The two methods alluded to for the discrimination of quadratic and non-quadratic residues, or, which is the same thing, for the determination of the value of the symbol (-), are not satisfactory, — the first because it supposes a reference to a Table of Indices (i.e. to a 50 EEPOET ON THE THEORY OF NUMBERS. [Art. 16. recorded solution of the problem it is proposed to solve), the second on account of its inapplicability to high numbers. A very different solution of the problem is supplied by a theorem which is known as ' Legendre's Law of Quadratic Reciprocity,' and which is, without question, the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss* ' the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. The theorem is as follows : — ' I£p and q be two uneven prime numbers, (|)=(_l)i(P-m,-.(|); (i); to which we must add the complementary propositions relating to the resi- dues — 1 and 2, In (ii), p is supposed to be positive ; in (i), p and q are supposed not to be simultaneously negative. The equation p (^) = ( " 1)^""" ''-'' may be expressed in words by saying that ' if ^ and g be two primes, the quadratic character of p in regard to q is the same as the quadratic character of 5 in regard to p ; except both p and q be of the form 4n + 3, in which case the two characters are opposite instead of identical.' Gauss, who attributes the first enunciation of this theorem to Legendre, while he justly claims the first demonstrati6n of it for himself f, appears to have considered that Euler was unacquainted with the theorem, at least in its simple * {Jacobi, Crelle, vol. xix. p. 314.} + <Pro primo hujus elegantissimi Theorematis inventore ill. Legendre absque dubio habendus est, postquam louge antea summi geometrae Euler et Lagrange plures ejus casus speciales jam per inductionem detexerant In ipsum theorema proprio marte incideram anno 1795, dum omnium, qusB in arithmetica sublimiori jam elaborata fuerant, penitus ignarus, et a subsidiis literanis omuino prsecluHUS essem. Sed per integrum annum me torsit, operamque enixissimam effugit, etc' — Comm. Soc. Qott. voL xvi. p. 69. Art. 16.] EEPORT ON THE THEORY OF NUMBERS. 57 form. (See Disq. Arith., Art. 151.) Nevertheless, we find in the ' Opuscula Analytica' of Euler, vol. i. p. 64, a memoir* the concluding paragraph of which contains a general and very elegant theorem, from which the Law of Reciprocity is immediately deducible, and which is, vice versd, deducible from that law. But Euler (loc. cit.) expressly observes that the theorem is undemonstrated ; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers ; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only ; though in one memoir he seems to have imagined (for his language is not very precise) that he had obtained a satisfactory demonstration. The theorem, in a form precisely equivalent to that in which we have cited it, was first given by Legendre, in a Memoir contained in the 'Histoire de I'Acaddmie des Sciences' for 1785. (See pp. 516, 517.) But the demonstration with which he has accompanied it is invalid for several reasons. (See Gauss, Disq. Arith., Arts. 151, 296, 297, and the Additamenta.) lAdditionf. Legendre's investigation of the law of reciprocity (as presented in the ' Thdorie des Nombres,' vol. i. p. 230, or in the ' Essai,' ed. 2, p. 198) is invalid only because it assumes, without a satisfactory proof, that if a be a given prime of the form 4n + l, a prime h of the form in + 3 can always be assigned, satisfying the equation (t)= —^- M. Kummer (in the Memoirs of the Academy of Berlin for 1859, pp. 19, 20) says that this postulate is easily deducible from the theorem demonstrated by Dirichlet, that every arithmetical progression, the terms of which have no common divisor, contains prime numbers. It would follow from this, that the demonstration of Legendre (which depends on a very elegant criterion for the resolubility or irresolubility of equations of the form ax^ + by'^ + cz^ = 0) must be regarded as rigorously exact (see, however, the 'Additamenta' to arts. 151, 296, 297 of the Disq. Arith.). In the introduction to the memoir to which we have just referred, the reader will find some valuable observations by M. Kummer on the principal investigations relating to laws of reciprocity.] * Observationes circa divisionem quadratorum per numeros primes (Comment. Arith. vol. i. p. 477). t Tlie additions to Arts. 16, 20, 22, 24, 25, 36, 37, and 38 were published at the end of Part II. of the Report (1860). I 58 REPORT ON THE THEORY OF NTTMBERS. [Art. 17. 17. Jacobi's extension of Legendre's Symbol. — The symbol (m, the introduc- tion of which has greatly contributed to simplify the theories of the higher arithmetic, does not appear in Legendre's Memoir of 1785. It first occurs in the ' Essai sur la Thdorie des Nombres ' ; the first edition of which appeared at Paris in 1798, and the second in 1808. Jacobi, in a note communicated to the Academy of Berlin in 1837*, has extended the notation of Legendre. If P=PiPiP3, ... where Pi, Pi, Ps denote (equal or unequal) uneven prime numbers, Jacobi defines the symbol yp) by the equation {^\ ^ {^\ th.\ (h\ \p)-\pj\pj ypj-' and observes that we then have the equations (|)=(-iM^^-^)(«-^>(|). (i); P and Q denoting any two uneven numbers relatively prime, the signs of which are subject to the same restrictions as the signs of ^ and q in the corresponding formula of Art. 16. The theorems expressed by these formulae of Jacobi are very easily deducible from the formulae of Legendre, and will be found in the Disq. Arith. (Art. 133). To prevent misconception, however, it is proper to observe that, while Legendre's equation (— ) = 1 is a necessary and sufficient condition for the resolubility of the congruence x" = k, mod p, Jacobi's equation \-p) = 1 f where P is not a prime number, though a necessary, is not a sufficient condition for the resolubility of the corresponding congruence x^ = k, mod P. That congruence requires for its resolubility that the conditions should separately be satisfied ; PuPi, ... denoting the unequal prime factors of P. Gauss (who had in the course of his own early researches arrived inde- * Ueber die Kreistheilung nnd ihre Anwendung auf die Zahlentheorie. See the Monats-Bericht of the Berlin Academy, vol. ii. p. 127 (Oct. 16, 1857), or Crelle's Journal, vol. xxx. p. 166, Art. 18,] REPORT ON THE THEORY OF NUMBERS. 59 pendently at the Law of Quadratic Reciprocity), before finally abandoning the theory, succeeded in obtaining no fewer than six demonstrations of this funda- mental proposition. The first two are contained in the Disq. Arith. (Arts. 125-145, and Art. 262) ; the third and fourth in two memoirs presented in 1808 to the Society of Gottingen (Comm. Soc. Gott. vol. xvi. p. 69, Jan. 15, and Comm. Recentiores, vol. i. Aug. 24), of which the latter bears the title ' Sum- matio serierum quarundam singularium.' The fifth and sixth appeared nine years later in the memoir entitled ' Theorematis Fundamentalis in doctrina de Residuis quadraticis demonstrationes et ampliationes novae ' (Comm. Recent, vol. iv. p. 3, Feb. 10, 1817). The fourth of these demonstrations is probably that which is promised in the Disq. Arith., Art. 151, but which does not appear in that work, because (as it woidd seem) Gauss had not yet succeeded in overcoming the diffi- culties connected with it. Independently of the fundamental importance of Legendre's Law of Reci- procity, these demonstrations of Gauss possess such intrinsic interest, and have contributed so much to the progress of the science, that we shall briefly review them here. 18. Gauss's First Demonstration. — -The first demonstration (Disq. Arith., Arts. 125-145), which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal (vol. xlvii. p. 139), and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished. Let X represent any uneven prime. The single observation that (|)=->-d) shows that the theorem of reciprocity is true for primes inferior to 7. To establish its universal truth, it is, consequently, sufficient to show that, if true for all primes up to X exclusively, it is also true for all primes up to X inclusively. Let the theorem therefore be assumed to be true for all primes inferior to X ; let p be any one of those primes ; and let the eight cases [2 x 2 x 2 = 8] be con- sidered separately, which arise from every possible combination of the hypotheses (a), (|)= -Kl, or = -1; (/3), X^l, or = 3, mod 4 ; iy),p = l, or =3, mod 4. It has to be shown that, in each of these eight cases, the symbol ( — J actually has the value which the Law of Reciprocity assigns to it. The nature of the T 2 I 60 REPORT ON THE THEORY OP NUMBERS. [Art. 18. proof in the four cases in which (?^)= +1 will be rendered intelligible by a single example. Let (y) - ^ ^"*^ ^®^ ^ =P = ^» ™*^ ^- ^y virtue of the sjrmbolic equation ^^) = 1, we can establish the congruence x^ =p, mod X, or (which is the same thing) the equation x^=p + \y; in which we may suppose x even and less than X, y positive, less than X and of the form 4w + 3. From this equation it appears that (^— ) = 1, and (—) = 1, the symbol (~j being here used with the meaning Jacobi has assigned to it. But every prime divisor of y is less than X ; and, therefore, by Jacobi's formula of reciprocity (which is valid for all uneven num- bers less than X, since by hypothesis Legendre's law is valid for all primes less th=u, X), (-p = (£) = !. But C-1) = . - (|) (i) ; so that, feally. (A) . , in conformity with Legendre's law. We have here assumed that x is prime to p; a. slight modification in the proof will adapt it to the contrary sup- position. Again, the two cases in which (y^ j = - 1, and X = 3, mod 4, admit of simi- lar treatment. For the equation ( y") = ~ 1 involves also the equation (-r^) = + 1 , because X = 3, mod 4. We have therefore the congruence x^ = —p, mod X, which will serve to replace the congruence x^=p, mod X, which presents itself in the four cases first mentioned. But the two remaining cases, in which (— )= —1, X = l, mod 4, require a different mode of treatment. By a singularly profound analysis. Gauss has succeeded in showing that every prime of the form 4 n. -f- 1 is a non-quadratic residue of some prime less than itself Assume, therefore, the existence of a prime w, less than X, and satisfying the condition (— ) = —1. This condition implies that (^) ^ ~ -^ ' ^'°'' ^ (t) "^^^® ®<1"^ ^^ + ^> ^^ should have (— ) = + I, by one of the first four cases. Hence we may infer that (^) = +1. and may establish the congruence x^ = isrp, mod X, which, treated as in the preceding cases, will lead us to the conclusion that (— ) (— ) = 1, i.e. that (-) = - 1. Art. 19.] REPORT ON THE THEORY OF NUMBERS, 61 19. Gauss s Second, Third, and Fifth Demonstrations. — The second demon- stration (Disq. Arith. 262) depends on the theory of quadratic forms, and will be refen'ed to in its proper place in this Report [see Art. 115]. The third and fifth (which are in principle very similar to one another) depend on much simpler considerations. A half -system of Residues for a prime modulus p is a system of \{p — '^) numbers r^, r^, ... ■?'i(j,_i), such that the p — 1 numbers +ri, +r2, ... +?'i(j,_]) constitute a system of residues prime to p. We might take for the numbers 7*1, r^, &c., the even numbers less than p (as Eisenstein has done : see Crelle's Journal, vol. xxviii. p. 246), but Gauss has preferred to take the numbers l,2,3,...l(p-l). Let q be any number prime to p, and let k be the number of the numbers, ?'*i> ?^2) 9^3) ••• 9^1 (p-i)) which are congruous, not to numbers in the series rj, rg, ... r^(p_i), but to numbers in the series — rj, —r^, ... — r^(p_i). It may be shown (by a method similar to that employed in Dirichlet's proof of Fermat's Theorem) that gi'"" i' = ( - 1 )^ mod p ; so that (?) = (- 1 )«^. ^- Hence if g be a prime as well as p, and k' denote the number which replaces k, when p and q are interchanged in the preceding considerations, we find that (f)(f ) = (-!)>-. It has, therefore, to be shown that k + k' = ^{p — l){q — l), mod 2. The way in which this is done is different in each of the two demonstrations, and is a little complicated in each of them ; but by the aid of a diagram the congruence may be demonstrated intuitively (compare Eisenstein: Crelle, xxviii. p. 246 {trans- lated by Cay ley in the Quart. Jour, of Math. vol. i. p. 186}). With a pair of axes Ox and Oy construct a system of unit-points in a plane : only let no such points be constructed on the axes themselves. If S be any geometrical figure, let (<S) stand for the number of unit-points contained inside it or on its contour. On Ox and Oy respectively take OA=\q, OB = ^p. Complete the parallelogram OACB, and draw its diagonals, OQC, AQB. It is then easily seen that * {Mr. Morgan Jenkins in a paper read to the London Mathematical Society [vol. ii. p. 29, 1867] shows that (-^) = (—1)*, Q and P uneven, and (^) being Jacobi's symbol.} 6S REPORT ON THE THEORY OF NUMBERS. [Art. 20. k = {QCA) - (QBO), k'=:{QBC) - (QOA), k+k'={ABC) - (AOB), = {OABC)- 2 {AOB), = (OABC), mod 2. But (OABC) = i(p - 1) (? - 1) ; therefore, finally, k + }^ = i{p-l)iq-l). mod2. These demonstrations (the Ist, 3rd, and 5th) introduce no heterogeneous elements into the inquiry (the geometrical method of the present article is to be regarded only as an abbreviation of an equivalent and purely arithmetical process) ; they are based on the principles of the two theories with which the Law of Reciprocity is most intimately connected, — those of the residues of powers, and of quadratic congruences. The third, in particular, appears to have commended itself above the rest to Gauss's judgment*. 20. Gauss's Fourth Demonstration. — The fourth and sixth demonstrations, though somewhat different from one another, are both intimately connected with the theory of the division of the circle. They must, therefore, be regarded as less direct than the earlier proofs, but they have contributed even more to the methods and resources of the higher arithmetic. The fourth depends on the formula H-r + r* + r« + ...+r(''-»>' = ii<"-"'V«..., (A) in which i represents (as throughout this Report) an imaginary square root of — 1 ; n is any uneven number, ^/n its positive square root, 2^ . . 2ir , i j^, . r = cos h t sm Let the series n n l+r' + r** + r'*+...4-r<»-i)'* be denoted by •^{k,n); in the particular case in which w is a prime number, it is easy to see that /k\ '<^(k,n) = (-)>lf(l,n). Further, p and q denoting two prime numbers, it is * ' Sed omnes hse demonstrationes,' (he is speaking, apparently, of the 1st, 2nd, 4th, and 6th,) 'etiamsi respectu rigoris nihil desiderandum relinquere videantur, e principiis nimis heterogeneis derivatse sunt ; prima forsan excepts, quae tamen per ratiocinia magis laboriosa procedit, opera- tionibusque proUxioribus premitur. Demonstrationem itaque genuinam haetenus baud affuisse non dubito pronunciare; esto jam penes peritos judicium, an ea, quam nuper detegere successit,' (the 3rd,) 'hoc nomine decorari mereatur.' — Comm. Soc. Gott. vol. xvi. p. 70. Art. 20.] REPORT ON THE THEORY OF NUMBERS. 63 found by actual multiplication of the two series >f'(|>, g") and "^(qjp) that + (i>,j)x+fep) = + (l.«); thatis (|)(^) = -JM_. If we substitute for the functions •v|' their values given by the equation (A), we find ,"n\ ^ n \ an equation which gives a relation between (— ) and (— ) coincident with that assigned in Legendre's Law of Reciprocity. The equation (A) is not easy to demonstrate. It is not indeed difficult to show that the sum of the series on the left-hand side is ± ^n when n=l, mod 4; and +i^n when n = 3, mod 4. But the determination of the ambiguous sign in these values appears to have long occupied Gauss. He has effected it in his memoir (the ' Summatio Serierum &c.' ) by establishing the equality l+r-|-r* + r9-|-...-|-r<»-"' = (r-r-i)(r3-r-3)...(r"-2-r-» + 2) ,,.^ (B) which he obtains by writing r for x, and n — 1 for m, in the series 1-a;"* (l-a;'")(l-a;"'-') _ (l-x'»)(l-a;"'-i)(l-a;'»-2) T^ "•" {l-x){l-x^) {l-x){l-x^){l-x^) ■•" •"' This series when m is a positive integer becomes an integral algebraical function, and is proved by Gauss to be zero if m be uneven ; and if m be even, to be equal to the product (1 — cc) (1 — a;^) ... (1 — x"*"'). From this last obseirvation, the demonstration of the formula (B) naturally flows. If n be an even number, the formula (A) becomes l+r + r* + r^ + ...+r^"-'^' = {l+i)^ or =0, (A') according as » is evenly or unevenly even. A very different, but a simpler demonstration of these formulae (A) and (A'), depending on the properties of the definite integrals cos x^ dx, I sin x^ dx, or / e"' dx, 00 «/ — QO ./— OO has been given by Dirichlet in his memoir, ' Application de 1' Analyse Infinit^si- male h. la Thdorie des Nombres' (CreUe, vol. xxi. p. 135). The same formulae have also been deduced by Cauchy from the equation ai(i + e-'" + e-«'^ + e-«'^ +...)= U{\ + e-^' ■\-e-'^ + e-''^ + ...), or ^ + e-a= + e-*'»'-l-e-^'^+... = ^ (i-|-e-*' + e-**^ + e-3'^+ ...). 64 REPORT ON THE THEORY OF NUMBERS. [Art. 20. in which ah = ir, {a* and 6* denoting real positive quantities, or imaginary quan- tities the real parts of which are positive ; the real parts of a, h have the same sign as a* 6* = ± ^v}*. This equation Cauchy obtained, as early as 1817, by the principles of his theory of reciprocal functions ; but it is also deducible from known elliptic formulae. (See a note by M. Lebesgue in Liouville's Journal, vol. v. p. 186. {See also Crelle, xvii. p. 57, and the Berlin Transac- tions for 1835 ; the former Memoir contains the criticism of M. Libri's proof}) If in it we write « • a^-tll for a\ and /S^ -J- *^ for h\ n 2 a and P being two evanescent quantities connected by the relation »o = 2i8, the two series ^o(i + e-a' + e-4a«^g-9a»4. _) and 2p{\-ve-^' + e-^^ + e-'^+...) become respectively y\r(l,n)xf°°e-'''dx, and (1 -|-e-J»'") x /""e"^' cZx; Jo Jo whence, dividing by the definite integral, and observing that a = a/ — e-^', ^ n we obtain finally, in accordance with the formulae of Gauss, ^(i.«)=^x/«(i+^')(i+e-*'"') = vA^^:^^t. For the case in which n is a prime number, the equality (B) has been * {Put a' = — iTTO) ; we get the formula in Lacroix, vol. ii. p. 408 ; and the condition that a is positive is the same as that the real part of V—ico is positive. See LiouvUle (II.) vol. iii. p. 30 for a general formula.} t See M. Cauchy's ' Memoire sur la Theorie des Nombres ' in the M^moires de ' I'Acad^mie de France, vol. xvii, notes ix, x, and xi. See also the Comptes Rendus for April 1840, or Liouville's Journal, vol. v. p. 154 ; and compare (besides the note of M. Lebesgue quoted in the text) a memoir by the same author in Liouville, vol. v. p. 42. {Writing a* = a'- ?^, b* = ^ + |^, I find>/^(m, n) = i a/— (l + ,)^(-n, 4m). If m = 1, this is right. If n = 4r, since ^( — 4v, 4m) = 4\lr(—v, m), we have ylf{m,4v)=2Ay~{i+i)yt'{-v,m), making the case when n is even depend on the case when n is uneven, and agreeing with Art. 104, note.} Art. 21.] REPORT ON THE THEORY OF NUMBERS. 65 established in a very simple manner by M. Cauchy* and M. Kroneckerf. But, as these latter methods have not been extended to the case in which n is a composite number, they cannot be used to replace Gauss's analysis in this demonstration of the law of reciprocity. From the formula (A) combined with the equation '^(k,p) = (-) ^O-^P)' p denoting a prime number, we may infer " -)\/P= 2 coss^ ; 2 sms^ P^ . = P 8 = P k\ /- 'e.P-^ . „2^7r »=P-1 2kTr or (_)^_p= ^ sms2 ; 2 coss^ according as ^ = 1, or =3, mod 4. These formulae serve to express the value of the symbol ( - j by means of a finite trigonometrical series, and are, therefore, of very great importance. Con- versely, the circumstance that a trigonometrical summation should depend on the quadratic characters of integral numbers, may serve of itself to show the use of abstract arithmetical speculations in other parts of analysis. \Additioii. Dirichlet's demonstration of the formulae (A) and (A') first appeared in Crelle's Journal, vol. xvii. p. 57. Some observations in this paper on a supposed proof of the same formulae by M. Libri (Crelle, vol. ix. p. 187) were inserted by M. Liouville in his Journal, vol. iii. p. 3, and gave rise to a controversy (in the Comptes Rendus, vol. x) between MM. Liouville and Libri. The concluding paragraphs of Dirichlet's paper contain the application of the formulae (A) and (A') to the law of reciprocity (Gauss's fourth demonstration).] ' 21. Gauss s Sixth Demonstration. — This demonstration depends on an investigation of certain properties of the algebraical function »=j)-2 8 = in which p is a prime number, j a primitive root of p, k any number prime to p, and X an absolutely indeterminate symbol. These properties are as follows : — \ —x'' (1) ^/-{- 1)J<P- "p is divisible by , (2) ^, - ^j is divisible by 1 - x^ * In the M^moire sur la Th6orie des Nombres, Note xi, or Liouville, vol. v. p. 161. t Liouville, New Series, vol. i. p. 392. K 66 REPORT ON THE THEORY OF NUMBERS. [Art. 21. (3) If ifc = <7 be a prime number, f ,« - ^, is divisible by q. From (1) we may infer that ^,'-' - ( - 1)*'""" ^i-\) ph(1-^.) is divisible by ^-^ ; and, by combining this inference with (1) and (2), we may conclude that is also divisible by - ; that is to say, (_i)i(i'-i)j9r(_i)j(j'-i)(9-«j3i<fl-i)-(2)l is the remainder left in the division of the function ^i (^Z - f ,) by - — — • But \. JO every term in that function is divisible by q ; the remainder is therefore itself divisible by q. We thus obtain the congruence ( _ l)|(p-i) (9-1) joi(?-i) = (ly mod q, which involves the equation (7)(f)-<-')""-""'""- Gauss has given a purely algebraical proof of the theorems (1), (2), and (3), on which this demonstration depends. The third is a simple consequence of the arithmetical property of the multinomial coefficient, already referred to in Art. 10 of this Report ; to establish the first two, it is sufficient to observe that ^t* - ( — l)i'''-'' j9 and ^t — (-)^i vanish, the first, if a; be any imaginary root, the second, if a; be any root whatever, of the equation x** — 1 = 0. If, for example, in 0_ O-ir the function ^j we put x = r = cos \-i sin — , we obtain the function -^{Icyp), which satisfies, as we have seen, the two equations [4'(^>i')? = (~^)^**~"i'' ^^^ >f'(A*,p) = f-)>|'(l,^). It is, indeed, simplest to suppose x = r throughout the whole demonstration, which is thus seen to depend wholly on the properties of the same trigonometrical function ■v/', which presents itself in the fourth demon- stration ; only it will be observed that here no necessity arises for the considera- tion of composite values of w in the function -^{k, n) ; nor for the determination of the ambiguous sign in the formula (A). In this specialized form. Gauss's sixth proof has been given by Jacobi (in the 3rd edit, of Legendre's ' Theorie des Art. 21.] REPORT ON THE THEORY OF NUMBERS. 67 Nombres,' vol. il. p. 391), Eisenstein (Crelle, vol. xxviii. p. 41), and Cauchy (Bulletin de Ferussac, Sept. 1829, and more fully M^m. de I'lnstitut, vol. xviii. p. 451, note iv. of the Mdmoire), quite independently of one another, but apparently without its being at the time perceived by any of those eminent geometers that they were closely following Gauss's method. (See Cauchy's Postscript at the end of the notes to his M^moire ; also a memoir by M. Lebesgue in Liouville, vol. xii. p. 457 ; and a foot-note by Jacobi, Crelle, vol. xxx. p. 172, with Eisenstein's reply to it, Crelle, vol. xxxv. p. 273.) MM. Lebesgue* and Eisenstein f have even exhibited a proof essentially the same in a purely arithmetical form, from which the root of unity again disap- pears, and is replaced by unity itself. Eisenstein considers the sum c'--2(|)(|)-0. in which k^, ki, ...kg denote q terms (equal or unequal) of a system of residues prime to p, the sign of summation extending to every combination of the numbers ki, k^, ...k^, that satisfies the congruential condition ki + k2 + k3+ ...+kg = a, mod^. This sum is, in fact, the coefficient of r" in the development of the qth power of ^=P-^^k\ . r*— 1 the function 2 ("")^> {reduced by the equation r^ — 1 = 0; not r = 0}, which is equivalent in value to Gauss's function ■^'(1,^). From the equation it follows that ^ f'^V- =(-l)i(i'-»)t«-i)«J(«-i)x 2 f-V; whence C„ = (- l)i'i'-»>(«-»(-)^J<«-').:t: • See Liouville's Journal, voL ii. p. 253, and vol. iii. p. 113. (The proof of the law of reciprocity will be found in sect. i. art. 5, and sect. iii. art. 2, of the memoir). See also the memoir referred to in the text, Liouville, vol. xii. p. 457. + Crelle's Journal, vol. xxvii. p. 322. t {This assumes that C^ = 0. Adding the p—l equations 2C^r' = (-l)*(p-i) («-i) ^i(»-i) 2(A)r'' to the equation 2/->Ca=0, we obtain C„=0 ; then the equation follows from the irreducibility of r— 1 ' K 2 68 REPORT ON THE THEORY OF NUMBERS. [Art. 22. And again, since r'rv*)r>]'H'i'-'(*)>*H(S)'r'(|)^.modj, we have the congruence ^ ^ /a\ /^y ^^^ But these results, which, taken together, establish the law of reciprocity, are obtained by Eisenstein from his arithmetical definition of (7„, without any reference to the trigonometrical function ^{l,p). If we write that function in the form 2 r*', instead of the form 2 ( - )r*, we obtam from its gth power the coefficient C"„ considered by M. Lebesgue. This coefficient, which is con- nected with Ca by the equation C"<,=p«"'-I-C„, represents the number of solu- tions of the congruence x^^ ^-x^^ + x^^ -{-... +Xq^ = a, mod q. From this definition M. Lebesgue deduces the equation C7'„ =_p«-» -I- ( - l)!*!*-" («-i) (-)p2<«-'\ and the congruence C",, = ! + (-) (-), mod q, by processes which, though different from those of Eisenstein, involve, like them, the consideration of integral num- bers only. 22. Other proofs of the Theorem of Reciprocity have been suggested to subsequent writers by a comparison of the different methods of Ga,uss. The x^ —\ symbol r denoting a root of the equation — = 0, it is very easily shown that (r_r-')'(r2-r-2)\..(ri<''-i>-r-i(*-'))' = (-l)2'''-i>p. (C) It is natural therefore to employ this equation to replace the equation which presents itself in the 4th and 6th methods of Gauss. It is also found that the product * - J (? - 1) r** — r- *« ,n^ n \, is equal to i^- (D) This is an immediate consequence of the property of a half-system of Residues (see Art. 19 supra) on which Gauss's 3rd and 5th methods depend. From a combination of the equations (C) and (D), the law of reciprocity is immediately deducible. (See a note by M. Liouville, Compt. Rend. vol. xxiv., or Liouville's Journal, voL xii. p. 95, and especially a memoir by Eisenstein, entitled ' Appli- [i-p — 1 .1.. -|2 Art. 23.] REPORT ON THE THEORY OF NUMBERS. 61^ cation de I'Algebre k rArithm^tique transcendante,' Crelle, vol, xxix. p. 177. The proof by the same author in vol. xxxv. p. 257, is the same as that in the earlier memoir, only that the properties of the circular functions, which here replace the roots of unity, are in the later memoir deduced immediately from the definition of the sine as the product of an infinite number of factors.) [Addition. From a general theorem of M. Kummer's (see Arts. 43, 44 of this Report), it appears that the congruence r^ = ( — l)z'^~'^A, mod q, is or is not resoluble, according as g2<^-i>s + 1, or = — 1, mod X, — a result which implies the theorem of quadratic reciprocity. This very simple demonstration (which is, however, only a transformation of Gauss's sixth) appears first to have occurred to M. Liouville (see a note by M. Lebesgue in the Comptes Rendus, vol. li. pp. 12, 13).] 23. Algorithm for the Determination of the Value of the Symbol (p) — Gauss has shown in the memoir ' Demonstrationes et ampliationes novae,' already quoted, that, if ^ be a prime number, the value of the symbol (—j may be Q . ^ obtained by developing the vulgar fraction — in a continued fraction, and con- P sidering the evenness or unevenness of a certain function of the quotients and remainders which present themselves in the development. Jacobi has observed (see Crelle, vol. xxx. p. 173) that a much simpler rule may be obtained, by the use of his extension of Legendre's symbol to the case when p is not a prime. The following is the form in which the rule has been exhibited by Eisenstein (see Crelle, vol. xxvii. p. 319). Jjet pi, p^ be two uneven numbers prime to one another, and let us form by division the series of equations Pi = ^hP2 +f2i>3, P2=^hP3 +f3Pi, Pfi = "kiiPft + i + efi + i, in which e^, e^, ... e^^j denote positive or negative units, and J9j, p2, Ps, ... which are all positive and uneven, form a descending series. Let o- denote the number of the quantities 2kr_ip^ + e^p^^i, in which both Pr and e^Pr^i are of the form 471 + 3 ; then (— ) = ( — 1)". The demonstration of this result flows immediately from the definition of Jacobi's symbol of reciprocity. A numerical example is added (see Disq. Arith., Art. 328) from which the to REPORT ON THE THEORY OF NUMBERS. [Art. 24. reader will perceive the utility of these researches in their practical application to congruences. Let the proposed congruence be x*" = - 286, mod 4272943, where 4272943 is a prime number. —286 We have to investigate the value of the symbol (— - — ), in which p is written for 4272943. Now (^) = (^)x(Px(f)=-(f)' because (^^) = - 1, and /-) = + 1, p being of the form 8»- 1. To find the value of ( ) , we have P 143 = X 4272943 + 143 f, 4272943 = 29880 X 143 + 103 t, 143 = 2x103-63, 103 = 2x63-23, 63 = 2x23 + 17, 23 = 2x17-11, 17 = 2xll-5t, 11 = 2x5 + 1. The obelisk (t) denotes that the equation to which it is affixed is one of those enumerated in a. Hence / 143 N , ,„ , , / -286 N (4272943) = (-^) =-^' ^^^ (4272943)=+^' or the proposed congruence is resoluble. Its roots (as determined by Gauss) are ±1493445. 24. Biquadratic Residues. — Reverting to the general theory alluded to in Art. 12, we see that, when ^ is a prime of the form 4/i + l, the congruence ar* — 1 = 0, mod p, admits four incongruous solutions ; these are +1, — 1, and the two roots of the congruence cc^ + l = 0, mod^, which we shall denote by +/and -/, or by/ and /^ so that the four roots of a5*-l = are 1, / -1, and/^ Further, if ^- be any number prime to p, k satisfies one or other of the four con- gruences — (i.) A;i<''-»)=1, mod^?. (iii.) ;fci'''-»= - 1, modjo. (iL) A;i<'-"=/, mod_p. (iv.) U^p-^i= f^,m(Ap. We see therefore that the p—1 residues of p divide themselves into four classes, comprising each \{p-l) numbers, according as they satisfy the 1st, 2nd, 3rd, or 4th of these congruences. The first class comprises those numbers a for Art. 24.] REPORT ON THE THEORY OF NUMBERS. 7| which the congruence x* = a, mod p, is resoluble ; that is, the biquadratic residues of p ; the third comprises those numbers which are quadratic, but not biquadratic, residues of p ; the second and fourth classes divide equally between them the non-quadratic residues. We owe to Gauss two memoirs* on the Theory of Biquadratic Residues, which, whUe themselves replete with results of great interest, are yet more remarkable for the impulse they have given to the study of arithmetic in a new direction. Gauss found by induction that a law of reciprocity (similar to that of Legendre) exists for biquadratic residues. But he also discovered that, to demonstrate or even to express this law, we must take into consideration the imaginary factors of which prime numbers of the form 4 n + 1 are composed. By thus introducing the conception of imaginary quantity into arithmetic, its domain, as Gauss observes, is indefinitely extended ; nor is this extension an arbitrary addition to the science, but is essential to the comprehension of many phenomena presented by real integral numbers themselves. Gauss's first memoir (besides the elementary theorems on the subject) con- tains a complete investigation of the biquadratic character of the number 2 with respect to any prime j3 = 4?i-|-l. The result arrived at is that if j> be resolved into the sum of an even and uneven square (a resolution which is always possible in one way, and one only), so that p = a^ + b^ (where we may suppose a and b taken with such signs that a = 1, mod 4; b = af, mod p), 2 belongs to the first, second, third, or fourth class, according as ^6 is of the form 4n, 4ft-|-l, in + 2, or 4 n. -f- 3. The considerations by which this conclusion is obtained are founded (see Art. 22 of the memoir) on the theory of the division of the circle, and we shall again have occasion to refer to them. In the second memoir Gauss developes the general theory already referred to, by which the determination of the biquadratic character of any residue of p may in every case be efiected. The equation p = a^ + b^ shows that p = {a + bi) (a — bi), or that p, being the product of two conjugate imaginary factors, is in a certain sense not a prime number. Gauss was thus led to introduce as modidus instead of p one of its imaginary factors : an innovation which necessitated the construction of an arithmetical theory of complex imaginary numbers of the form A + Bi. The • Theoria Residuorum Biquadraticorum. Commentatio prima et secunda. (Gottingse, 1828 and 1832, and in the Coram. Recent. Soc. Gott., vol. vi. p. 27 and vol. vii. p. 89.) The articles in the two memoirs are numbered continuously. The dates of presentation to the Society are April 5, 1825, and April 15, 1831. 78 REPORT ON THE THEORY OF NUMBERS, [Art. 25. elementary principles of this theory are contained in the memoir in question ; they have also been developed by Lejeune Dirichlet with great clearness and simplicity in vol. xxiv. of Crelle's Journal (pp. 295-319, sect 1-9)*. The following is an outline of the definitions and theorems which serve to constitute this new part of arithmetic. [Addition. A note of Dirichlet's, in Crelle, vol. Ivii. p. 187, contains an elementary demonstration of Gauss's criterion for the biquadratic character of 2. From the equation p = a^ + h^, we have (a + by = 2 ah, mod^, and hence (a + h)i^'- » = 2^(»'-»' a*<^-^' &i^*-" = (2/)**"-" ai<»-i>, or, which is the same thing, Cf>W'-"{p (A) But (-) = (~) = 1, hec&use p = ¥, mod a ; and ( ■) = (-"^). or, observing that 2p = (a + bY + (a-by, since /^ + 1 =0, mod^. Substituting these values in the equation (A), we find 2i(i>-i) =J'iah^ mod p, which is in fact Gauss's criterion.] 25. Theory of Complex Numbers. — The product of a number a + hi by its conjugate a — hi is called its norm ; so that the norm of a + hi is a^ + b- ; the norm of a (which is its own conjugate) is a*. This is expressed by writing N{a + hi) = N{a — hi) = a^ + b^; N(a) = a^. If a and ^ be two complex numbers, we have evidently N(a) x N(0) — N{a/3). There are in this theory four units, 1, i, —1, —i, which have each of them a positive unit for their norm. The four numbers a + hi, ia — h, —a — ih, — ia + h (which are obtained by multiplying any one of them by the four units in succession, and which consequently stand to one another in a relation similar to that of + a and — a in the real theory) are said to be associated numbers. These four associated numbers with the numbers respectively conjugate to them form a group of eight numbers (in general * The death of this eminent geometer in the present year (May 5, 1859) is an irreparable loss to the science of arithmetic. His original investigations have probably contributed more to its atlvancement than those of any other writer since the time of Gauss; if, at least, we estimate results i-ather by their importance than by their number. He has also applied himself (in several of his memoirs) to give an elementary character to arithmetical theories which, as they appear in the work of Gauss, are tedious and obscure; and he has thus done much to popularize the theory of numbers among mathematicians — a service which it is impossible to appreciate too highly. I Art. 25.] REPORT ON THE THEORY OF NUMBERS. 78 different), all of which have the same norm. These definitions are applicable whatever be the nature of the real quantities a and h. If a and h are both rational, the complex number is said to be rational ; if they are both integers, a + bi is a complex integral number. One complex integer a is said to be divisible by another /3, when a third y can be found such that a = ^y. Adopting these definitions, we can show that Euclid's process for investigating the greatest common divisor of two numbers is equally applicable to complex numbers ; for it may be proved that, when we divide one complex number by another, we may always so choose the quotient as to render the norm of the remainder not greater than one-half of the norm of the divisor*. If, therefore, we apply Euclid's process for finding the greatest common divisor to two complex numbers, we shall obtain remainders with norms continually less and less, thus at last arriving at a remainder equal to zero ; and the last divisor will be, as in common arithmetic, the greatest common divisor of the two complex numbers. Similarly the fundamental propositions deducible in the case of ordinary integers from Euclid's theory are equally deducible from the correspond- ing process in the case of complex integral numbers. Thus, ' if a complex number be prime to each of two complex numbers, it is prime to their product.' ' If a complex number divide the product of two factors, and be prime to one of them, it must divide the other.' 'The equation ax — by=l, where a and b are complex numbers prime to one another, is always resoluble with complex numbers x and y, and admits an infinite number of solutions,' &c. A prime complex number is one which admits no divisors besides itself, its associates, and the four units. There are three distinct classes of primes in the complex theory : — 1. Ileal prime numbers of the form in + 3 (with their associates). 2. Those complex numbers whose norms are real primes of the form in + l. 3. The number 1 + i and its associates, the norm of which is 2. Instead of dividing numbers into even and uneven, we must here divide them into three classes, uneven, semi-even, and even, according as they are (1) not divisible by (l + i); (2) divisible by 1 -f- i, but not by (1 -I- i)^ ; (3) divisible by (1 +t)^ = 2i, or, which is the same thing, by 2. ^ „. a + bi ac + bd be—ad. .,,,.,, , ^ . ac + bd . , * Since ,. = -= — z^ H — „ — is i; u 7) be the inteffral number nearest to -^ — tj, and q that e + di e^ + d^ c' + d^ c^ + d' , ^ he— ad . , . . , nearest to , „ , }> + qi is the quotient required. 74 REPORT ON THE THEORY OF NUMBEBS. [Art. 25. Of four associated uneven numbers, there is always one, and only one, such that b is even and a + h-1 evenly even. This is distinguished from the others as primary. Thus — 7 and — 5 + 2 1 are primary numbers. A primary number is congruous to + 1 for the modulus 2(1 +i); whence it appears that the product of any number of primary numbers is itself a primary number. The conjugate of a primary is also primary. In speaking of uneven numbers, unless the contrary is expressed, we. shall suppose them to be primary. This definition of a primary number is that adopted by Gauss {I.e. Art. 36), and after him by Eisenstein, and we shall adhere to it in this Report. But Gauss has also suggested a second definition (which is for some purposes slightly more convenient), and which has been adopted by Dirichlet, who defines a primary uneven number to be one in which b is even, and a = 1, mod 4. The object of singling out one of the four associated numbers is merely that it serves to give definiteness to many theorems. For example, the theorem that 'every real number may be expressed as the product of powers of real primes in one way, and in one only,' may be now transferred in an equally definite form to the complex theory, ' Every complex number can be expressed in one way only in the form i'»(H-i)".^».£^CT..., where m, n, a, /3, y, &c. are real integral numbers. A, B, C ... primary com- plex primes.' If a + bi be a complex number, and N = N{a + bi) = a^ + b^, and if /t be the greatest common divisor of a and b, it can be shown that every number is con- gruous, for the modulus a + bi, to one, and one only, of the numbers x + iy, where x = 0,l,2,...:|^-l; y = 0,l,2,...h-l. These numbers therefore (or any set of numbers congruous to them) form a complete system of residues for the modulus a + bi. The number of the numbers X -I- iy is evidently N, so that the norm of the modulus represents the number of residues in a complete system. In particular, therefore, if the modulus a -f- 6* be a prime of the second kind, having p for its norm, the numbers 0, 1, 2, ...p — 1 represent a complete system of residues ; and if the modulus be a prime of the first kind, as q, the numbers included in the formula x + iy, where x and y may have any values from to ^ — 1 inclusive, will represent a complete system of residues. [Addition. Although the second definition has been adopted by Dirichlet in his memoir in Crelle's Journal, vol. xxiv (see p. 301), yet in the memoir Art. 27.] EEPORT ON THE THEORY OF NUMBERS. 75 ' Untersuchungen liber die complexen Zahlen' (see the Berlin Memoirs for 1841), sect. 1, he has preferred to follow Gauss.] 26. Fermat's Theorem for Co^nplex Numbers. — Dirichlet's proof of this theorem for ordinary integers is equally applicable to complex numbers, and leads us to the following result : — ' If j9 be a prime in the complex theory, and h any complex number not divisible by p, then k^P-'^ = 1, mod p.' Again, the demonstration of the theorem of Lagrange (see Art. 11) is equally applicable here (see Gauss, Theor. Res. Biq., Art. 50), and therefore the general theorems mentioned in Art. 12 may be extended, mutatis mutandis, to the com- plex theory. In particular, the number of primitive roots will be '^\_N{p) — 1], or the number of numbers less than N{p) — 1, and prime to it. It will follow from this that, if the modulus be an imxiginary prime p, eYery primitive root of Np in the real theory will be a primitive root both of p and its conjugate. Those Tables of Indices, therefore, in the ' Canon Arithmeticus,' which refer to primes of the form 4 n + 1 will continue to hold, if for the real modules we substitute either of the imaginary factors of which they are composed. For primes of the form 4?i-f-3 (considered as modules in the complex theory), it would be requisite to construct new tables, — a labour which no one as yet appears to have undertaken. 27. Law of Quadratic Reciprocity for Complex Numbers. — If ^ and q be any two uneven primes (not necessarily primary, but subject to the condition that their imaginary parts are even), and if we denote by M- the unit-residue of the power ^K^?-i], mod q ; so that — = -|- 1, or = — 1, according as ^ is or is not a quadratic residue of q : then a law of reciprocity exists, which is expressed by the equation Ul = -i . K p and q are both real primes, it is easily seen that either of them is a quadratic residue of the other in the complex theory, or M- = M^ = 1. But, as p may or may not be a quadratic residue of q in the theory of real integers, we see that the values of the symbols ~ and (—j are not neces- sarily identical. This theorem is only enunciated in Gauss's memoir (Art. 60), and, as he speaks of it as a special case of the corresponding theorem for biquadratic L 2 76 REPORT ON THE THEORY OF NUMBERS. [Art. 28. residues, it is probable that his demonstration of it was of the same nature with that which he had found of the law of biquadratic reciprocity. How- ever, a simple proof of it, depending on Legendre's law of reciprocity, has been given by Dirichlet in Crelle's Journal *. He shows that, if g be a prime of the first kind, f^^^^l = ( ° ^ )> ^^^ ^^^^' if a + &i be any prime of the second kind in which b is even, [^^] = O + foO ' '^^^ ^^^ ^^ reciprocity is easily deducible from these transformations. If, for example, a + U, a + fii, be primes of the second species in which both 6 and ^ are even, we have simultaneously ra + /8i-| _ y' aa + b^ \ r a + hi n ^ / aa + h^ \ la + bi}~\l) /' La-|-/8iJ~v ^ )' aa + 6/3>. where p = N{a + b{); i!T = N{a+ fii). But ( "°"^ ^ ) = (^^f^) . ^y Jacobi's formula (see Art. 17 supra) ; and ( j = ( — TX«}" ^^^^ p^ = (aa + 6|8)2 + (ttiS - bay ; whence we infer / pa- \ . i ■ i, • xu au- ( ^ , ^ 1 = 1, or, which IS the same thmg, \aa + b3y ^ -a + bi' The complementary theorems which have to be united with this formula are L-^]=(-)"-^ L^]=(-)' (see Dirichlet, Crelle, voL xxx. p. 312) ; and they, as well as the formula of reciprocity itself, admit of an extension similar to that which Jacobi has given to the corresponding formulae of Legendre. 28. Reciprocity of Biquadratic Residues. — We now come to the theorem which first suggested the introduction of complex numbers. If p be any (complex) prime, and h be any residue not divisible by p, we denote by C—\ the power i' of i, which satisfies the congruencb ^U^i"-!) = i. It * Crelle, vol. ix. p. 379. Art. 28.] EEPORT ON THE THEORY OF NUMBERS. W will be observed that when jp is a prime of the second species, the quadripartite classification of the real residues of p which we thus obtain is identical with that which we obtain for Np in the real theory (see Art. 24 supra) ; for the numbers y and —/ being the roots of the congruence cc^ + 1 =0, modiVj?, satisfy the same congruence for either of the complex factors of Np, and are therefore congruous to +i and —i, for one of those factors, and to —i and +i for the /lc\ other. Admitting this definition of the symbol ( — ) , Gauss's law of biquadratic reciprocity is expressed by the equation a and j8 denoting two primary uneven primes, and A and B being their norms. The complementary theorems relating to the unit i and the semi-even prime 1 + i are (ii.) (-J^ =i-|(a-l). (iii.) (l±l-\ =iU(a + «')-(l + <^)^), ^ ' \a-\-iah ^ ' Vt + ia'A ' in which a -Via denotes a primary uneven prime. These formulae, like those of the last article, are susceptible of the same generalization which Jacobi has applied to Legendre's symbol ; and we may suppose in the first that a and jS are any two primary uneven numbers, prime to one another ; and in the second and third that a + ia' is any primary uneven number. K, in the formula (i.) which expresses the law of reciprocity, a = a-via', fi = h + ih', it may be easily seen that the unit ( — l)?'^-!)?*^-!) is equal to ( — l)i(<»-i)-5(*-i). This gives us a second expression of the theorem. (See Eisen- stein, 'Math. Abhandl.' p. 137, or CreUe, vol. xxx. p. 193.) Further, if we observe that every primary number is either = 1, mod 4, or else = 3 + 2i, mod 4 ; and that ^(^ - 1) . :^(i? - 1) and i(a - 1) • i (^ "" ^) ^^® ^^^'^ numbers, unless both a and /3 satisfy the latter congruence, we may enunciate the law of biquadratic reciprocity by saying^- ' The biquadratic characters of two primary uneven prime numbers with respect to one another are identical, if either of the primes be =1, mod 4 ; but if neither of them satisfy that congruence, the two biquadratic characters are opposite.' This theorem is only enunciated by Gauss, who never published his demon- stration of it. ' Non obstante,' he observes, ' summ^ huius theorematis sim- plicitate ipsius demonstratio inter mysteria arithmeticae sublimioris maxime 78 REPORT ON THE THEORY OF NUMBERS. [Art. 29. recondita referenda est, ita ut, saltern ut nunc res est, per subtilissimas tantum modo investigationes enodari possit, quae limites praesentis commentation Ls longe transgrederentur.' — Theor. Res. Biq. Art. 67. Soon after the publication of the theorem, its demonstration was obtained by Jacobi, and communicated by him to his pupils in his lectures at Konigsberg in the winter of 1836-37 (see his note to the Berlin Academy, already cited in Art. 17). These lectures have unfortunately never been published ; but Jacobi's demonstration, from his criticism (see ibid.) on the first of those given ten yeai-s later by Eisenstein, appears to have been very similar to it. It is to Eisenstein that we are indebted for the only published proofs of the theorem in question. That great geometer (so early lost to arithmetical science — a victim, it is said, to his devotion to his favourite pursuit) has left us as many as five demonstrations of it ; the two earlier based on the theory of the division of the circle ; the last three, on that of the lemniscate. We proceed to explain the principles on which each of these two classes of proofs depends : — 29. Biquadratic Residues — Researches of Eisenstein. — It is possible, as we have seen, to obtain a proof of Legendre's law of Reciprocity by considem- tions relating to the function 2 (—jod', p denoting a real prime, and x a a;* — 1 root of the equation — = 0. This function is a particular case of the well- •*/ ■"" -I- known function (Introduced by Gauss and Lagrange into the theory of the division of the circle) F (6, a;) = 2 d' xt', where 6 is any root of the equation »=o ©"-'-1 . . . — ^—^j — = 0, 7 a primitive root of the congruence cc''"^ = 1, rciodp, and x a root x^ — 1 of the equation T~^- ^^ ^^^ quadratic theory we assign to 6 the value - 1 ; in the theory of Biquadratic Residues we put 6 = i, and are thus led to «=p-2 consider another particular form of the same function, viz. F(i, x) = 2 i' a^', p denoting a prime of the form 4»+ 1. 30. The function F (0, x) or F{6) is characterized by the following general properties; which have been given by Jacobi, Cauchy, and Eisenstein. (See Jacobi, Crelle, vol. xxx. p. 166 ; Cauchy, Mdmoire sur la Theorie des Nombres in the Mdm. de I'Acad. de I'lnstitut de France, vol. xvlii ; Eisenstein, Crelle, vol. xxvii. p. 269. {Also M. Lebesgue, Liouville, vol. xix}.) Art. 30.] REPORT ON THE THEORY OF NUMBERS. 7^ I. F (e, a^) = e-'^^^y^F (e, x), II. F(e)F(e-') = ei^p-^^p, F{e-'^)F{d-^) where -^ {&) does not involve x, and Is an integral function of 6 with integral coefficients '"'. The function -^ (9) satisfies the equation IV. ^^(0)^/' (0-0=1'. xP-i — 1 Lastly, let be a primitive root of — = 0, and in the function *}u ^ J. ^^^ i?'(0-'»-») let y be written for ; then if m and n be positive and less than p — 1, Tim denoting the continued product 1.2.3 ...m. Applying these equations to the particular form of the function F which we have to consider here, we find F{i)F(-{) = ii<p-'^p, ^{t) = l^}lM^ if 0i(p-i) = i, and m = 7i = |(p-l). {F(j)Y =p[^{i)J, ^|,(^i(p-l))=0, modj9. Let ■v|' (i) = a 4- &^ = ^1 ; >f' ( — ^) = a — 6^ = p2, so that p^ p^ =p. The congruence x|,[Yi(i.-i)] = 0, mod^, or a + 67?'^-i) = 0, modp, involves also the congruence a + 67i'*-'' = 0, mod^i; i.e. 7^^''-^'=^, mod^i ; so that T— j =i*. Hence we have, putting y'=k, mod^, i=i ^Pi^i * In this equation ^™ and ^" are supposed not to be reciprocals. 80 REPORT ON THE THEORY OP NTJMBERS. [Art. 30. From these formulae two cases of the law of Reciprocity are directly deducible. a. Let 5 be a real prime of the form 4» + 3. Raising S to the power q, we have S'=T\^)'^=T\^)'^.(^) r.modg. by (I,). Multiplying by S, we find or, observing that^a=jpA mod q, and p=PiP2, that is to say (j\-Q\ '^-^ which is in accordance with the law of Reciprocity. /8. Again, let gr be a prime of the form 4n + 1 ; then S" = (^)^S, mod q ; that is, -S'-^ = (^) , mod q, or pi(«-''jPiJ(«-i)=(-^) , modg; whence, if, = «,,„ (^) (|!)' = (i)" . But, by changing i into —i, (^y=(^), and (^y=(^), so that C^) = fi:) (B.) \q\ ^p\ The symbolic equations (A.) and (B.) lead immediately to the conclusion that if a and h be any two primary uneven numbers, one, at least, of which is real, we have (r) = (-) ; and that if a and h be both real, the common value of these symbols is + 1. By combining with these results the supplementary equation ( ,• n = i~i'"~^\ in which a-\-ia' denotes any primaiy uneven number, and also the self-evident equations, Art. 32.] REPORT ON THE THEORY OF NUMBERS. BX c(a±hi) = (ac + hd) ± bi(c ± di), a(c± di) = (ac + hd) + di(a + hi), /■a + hi\ /a — hi c-'rdiK ^c — di^^ ' c-i-di-' Eisenstein* investigates a relation between the symbols ( -pj and =-. ) , which, when a + hi and c + di are primary, coincides with that ex- a + bi'^i ^ '' pressed by the law of reciprocity. 31. The proof in Eisenstein's second memoir f is identical in its essential character with that in the first ; but he has given it a purely arithmetical form, independent of the theory of the division of the circle. Instead of the sum *=;>-! -y(;v ... . x^ — 1 S = 2 ( — ) 0^, in which a; is a root of the equation — = 0, he considers fe = l Vp/4 ^ x-1 the powers of the series 2 ( — ) , and arrives by a process purely arithmetical at the equations (A.) and (B.) of the preceding article. Thus the two forms in which he has exhibited his demonstration are precisely analogous to the two expressions which he has given to Gauss's sixth demonstration of Legendre's law (see above. Art. 21). 32. The proofs of the Law of Biquadratic Reciprocity, which are taken from the theory of elliptic functions, no less than those which we have just considered, depend in great measure on a generalization of the principles introduced by Gauss into his demonstrations of Legendre's law. Indeed, Gauss himself tells usj that his object in multiplying demonstrations of Legendre's law, was that he might at last discover principles equally applicable to the Biquadratic Theo- rem. It would be interesting to know whether the proof which he ultimately obtained of this theorem depended only on the division of the circle, or on elliptic transcendents. Jacobi appears to have believed the latter; for he expresses his opinion that his own demonstration of the Biquadratic Theorem See the memoir entitled ' Lois de Reciprocity,' in Crelle, vol. xxviii. pp. 53-67. t ' Einfacher Beweiss und Verallgemeinerung des Fundamental-Theorems fiir die biquadratischen Reate,' in Crelle, vol. xxviii, p. 223. X See the memoir, ' Theorematis Fundamentalis Demonstrationes et Ampliationes Novse,' p. 4 : ' Hoc ipsura incitamentum erat ut demonstrationibus jam cognitis circa residua quadratica alias aliasque addere tantopere studerem, spe fultus, ut ex multia methodis diversis una vel altera ad illustrandum argumentum afiine aliquid conferre posset.' ■ ■ g2 REPORT ON THE THEORY OF NUMBERS. [Art. 33. was widely different from that of Gauss* ; and he further conjectures that what induced Gauss to introduce complex numbers, as modules, into the theory of numbers, was not the study of any purely arithmetical question, but that of y' dx /(l_^\ f- 'This opinion of Jacobi's will not appear improbable, when we remember that in the • Disquisitiones Arithmeticae ' (Art. 335) Gauss promises an ' amplum opus ' on these transcendents ; and that a casual remark of his in relation to them renders it perfectly certain (as Dirichlet has observed) J that he was at that early period in possession of the principle of the double periodicity of elliptic functions — thus anticipating by twenty-five years the discoveries of Abel and Jacobi. Never- theless the close analogy we have endeavoured to point out between Gauss's sixth proof of the quadratic theorem, and the trigonometric demonstration of the biquadratic one, may perhaps incline us to the opposite opinion. Nor is the introduction of complex numbers, as modules, an idea unlikely to have suggested itself, when once complex numbers were admitted ; though it is remarkable that Jacobi, in the first printed memoir in which complex numbers appear, and to which we shall presently refer, seems not to have thought of this extension of his theory J. 33. Application of the Lemniscate Functions to the Biquadratic Theorem^. — Let 2>i be a complex prime (real or imaginary), p its norm ; and let the p—1 residues, prime to ^J^, be divided into four groups of 4(^ — 1) terms, after the following scheme : — (0) ^1, n, n(p-i)' (1) iVi, iVg, *>i(p-i), (2) - r„ -7-2, - ^(p-i), (3) -iri, -ir^, -*»'i(p-i), • ' Ueber die Kreistheilung,' Crelle, vol. xxx. p. 171. t Crelle, vol. xix. p. 314, or in the ' Monatsbericht ' of the Berlin Academy for May 16, 1839. t lu hifl ' Qedachtnissrede uber Karl Gustav Jacob Jacobi,' Mem. de I'Acad^mie de Berlin, 1852. This remarkable 61oge ie also inserted in Crelle's Journal, vol. lii, and in a French translation in Liouville's Joamal, vol. ii, 2nd series. § {Gauss's demonstration seems after all to have been more nearly comparable to the second and fifth of the Quadratic Theorems. See his Works, vol. ii : but I have not yet examined the paper carefully. } II See Eisenstein's memoir, ' Applications de I'Algfebre k I'Arithmetique transcendante,' in Crelle's Journal, vol. xxx. p. 189, or in Eisenstein's ' Mathematische Abhandlungen,' p. 121. Art. 33.] BEPORT ON THE THEORY OF NUMBERS. 83 so that of any four associated numbers one, and only one, appears in each group. Let ji be any residue prime to ^j ; h^, h^, h^, ... the nmnbers of the residues 2in, 2i^2, 21^(^-1) which belong to the groups (1), (2), (3), respectively; then 2ji(P-i)=i*i + 2fcj+3fc3^ mod^i. or 3i, (See Gauss, Theor. Kes. Biq., Art. 71.) The expression on the right-hand side of this equation may now be trans- formed by means of the Lemniscate function ^, defined by the equations The function (f> (v) is doubly periodic, the arguments of the periods being 2 a) and (1 +i) w, or, more simply, (1 +i) m and (1 —i) w, where dx _ /•! dx ,0 v(i-^) so that we have <j>(v + 2kw) = (p{v), k denoting any complex integer whatever. From this it appears that the relation of the Lemniscate functions to the theory of complex numbers is the same as the relation of circular functions to the arithmetic of real integers. The function <p (v) also satisfies the equation * ^ (z* v) = i* (p (v), whence ifc, + 2/.,+ 3/.3= l_£i_J.= (iL), (1) ^\ Pi / the sign of multiplication 11 extending to every residue r included in the group (0). Similarly, i£ qi, like pi,he a. prime, e)=-!^ (^) s denoting the general term of a quarter-system of Residues for the modulus qi. * {This, which is evident from the definition of (j) («), is also readily verified by applying the transformation oi = — ; we find the multiplier -Tp = i, &c.} — 1+0)' '^ M ' ' M 2 84 REPORT ON THE THEORY OF NUMBERS. [Art. 34. By an elementary* theorem in the calculus of Elliptic Fimctions, ^Vr is for every uneven value of A; a rational and fractional function o£ x = (f) (v). If —-), we have, by the principles of that calculus, 4>iPiV) _ Tl(x*-a*) <p{v) ~ Il(l-a«a:*)' ^ ' the sign 11 extending to all the different values of a ; and similarly, (l>{v) II(l-/3*x«)' ^ ^ ^ A = ^ ( — -)• Combining the equations (3) and (4) with (1) and (2), we find \qj, n{l-a*8*)' the sign of multiplication extending to the i(i> — 1)(3 — 1) combinations of the values of a and j8 ; whence, evidently, (Sl) (Pi) =(_i)i(i>-i)(«-i). The priority of Eisenstein in this singularly beautiful investigation is in- disputable. 34. In a later memoir ('Beitrage zur Theorie der Elliptischen Functionen,' Crelle, xxx. p. 185, or Math. Abhandl. p. 129), Eisenstein has put this proof into a slightly different form. He shows, by a peculiar method, that if ^ be an imaginary and primary complex prime, every coefficient in IT (a;* — a*) except the first is divisible by pi, and that for every primary uneven value of p, (whether prime or not) the last coefficient is pi, so that ( — l)i'''^~^^pi = Tla*. Representing therefore by p^ an imaginary and primary prime, by qi any complex prime, the equation (Pi) = n'-±J^ = u Vo,/ /2.s<o\ S') * {It is not the ordinary theorem of multiplication, for k is complex. Doubtless equations (3) and (4) may be immediately proved by the general method of comparing the zeros and infinities of either side.) Art. 35.] REPORT ON THE TKEORY OF NUMBERS. )6$ assumes the form U2\ = (^_ i^Hp-i)i.q-i) q^iiP-i)^ modp, or which establishes the law of Reciprocity for eveiy case except that of two real primes, when the value of the symbols (— ) = ( ) = 1 is at once apparent from their definition and from Format's Theorem. 35. A third, and no less interesting application of the theory of elliptic functions to the formula of Biquadratic Reciprocity, occurs in the memoir, ' Genaue TJntersuchung der Unendlichen Doppel-Producte, aus welchen die Elliptische Functionen als Quotienten zusammengesetzt sind' (Mathematische Abhandl. p. 213, or Crelle's Journal, vol. xxxv. p. 249). The function n= +00 •»= +00 irj/. ^ „=_oo m — 00 ^ m + nO' tx which is considered in this memoir, and in which the factor 1 — ^ is to be replaced by tx, coincides (if we disregard a constant factor) with the numerator of <l>{v), when that function is expressed as the quotient of one infinitely con- tinued product divided by another. This may be seen by comparing F{x) with the expression of the general elliptic function (p (a) given by Abel, viz. ;l = oo (i>(a)=aU (l + -^—^ n (\--^\ m = oo /!= 00 X n n «n = l /« = 1 1 + 1 + (a — mti))- 1 + (M-i) (M-i)^ 1 + (m-4)W 1 + (See Abel, (Euvres, vol. i. p. 213, equat. 178.) If we particularize this expression, by putting w = w (which changes ^ (a) into the Lemniscate-function) and then write wtx for a, we shall find that the iunction of x which appears in the numerator is precisely Eisenstein's func- tion F (x). This function (which is, consequently, a particular case of Jacobi's function H in his ' Fundamenta Nova ') is only singly periodic ; so that F{x) = F(x+ ~\ if n denote any real integer ; but F (x + -y^ is equal to the product of F(x) by an exponential function, if m be an imaginary complex 86 REPORT ON THE THEORY OF NUMBERS. [Art. 36. (Ui) '''U.F(x + —) number. (Compare the formulae of sect. 61 of the ' Fundamenta Nova.') The difficulty occasioned by this imperfect periodicity of F{x) Eisenstein has over- come by the introduction of the number t, which is supposed to represent a real even indeterminate integer. The formulae on which his proof depends, are (i) F(x + k)= e"''F{x), (ii) F{ix) =ie^''F{x), The symbol w which depends on x, but is independent of t, is different in each of these equations : m the first, k is any complex integer ; in the third, c is a numerical constant independent of x and p^; pi a primary number prime tot; p its norm ; and r the general term of the ^- 1 residues of p^, the sign of multiplication TI extending to every value of r. These equations, the first two of which depend on the most elementary properties of the function F{x) or H (see ' Fundamenta Nova,' loc. cit), while the third is of a more abstruse character, Eisenstein has established by methods which are peculiar to himself, and which it would take us too far from our present subject to describe. They serve to replace the formulae (j)(v)^(i){v + 2ku)); (})(iv) = i(j){v); (t>{piv) _ n(a:*-a«) (j){v) ~ U{l-a*x*) in Eisenstein's earlier demonstration ; and lead to the conclusion w still denoting some quantity independent of t. And since in this formula t may have any even value prime to pi and q^, it is impossible that c'" should have any value but that of one of the fourth roots of unity, so that we have e"'' = 1 ; which gives the law of Reciprocity. 36. An algorithm has been given by Eisenstein * for calculating the value of the symbol (j- — rr-rj by means of the development of , ..; - , in a continued * Crelle's Journal, vol. xxviii. p. 243. But the first invention of this algorithm, and of the similar one which exists in the Theory of Cubic Besidues, is due to Jacobi. (See the note, ' Ueber die Kreistheilung &c.,' so often cited in this report.) Art. 36.] BEPORT ON THE THEORY OF NUMBERS. 8T> fi^ction. This algorithm, in a slightly simplified form, is as follows : — Let a + ia = po, b + ib' = pi, and form the series of equations Po = hPi +i''^-P2, The numbers p^ and p^ are supposed to be uneven, and prime to one another ; Pi is primary ; the quotients k^, \, ki, ... k^ are all divisible by 1 +i, and are so chosen that the norms of ^^a. Pi,--- form a continually decreasing series (as is always possible) ; lastly, the units i'' are so chosen as to render pg, p^,... primary. Let jpg = a, + ia, ; let —\{ag—l) = d„ mod 4; and in the series Q^, 6^, ... 0„ + i, let p be the number of sequences of uneven terms. Then (— ) = i^f^'^^i*. ^Pi i Eacample. Let it be required to determine whether the congruence x*= -3381, mod 11981 be possible or impossible. Since 11981 = 109^ + 10'^, and is a prime number, the resolubUity of this congruence depends on that of the congruence a;*= —3381, mod ( — 109 + lOi). — ^^m v We have therefore to investigate the value of the symbol ( — —^ — TTT/ ' '^^^^ gives us the series of equations - 3381 = (31 + 3i) ( - 109 + 10^) + i» ( - 17 + 28^■), - 109 + 10i = ( 2 + 2i")(- 17 + 28i) + i''(-19-12i), - 17 + 28i= -2i (- 19-12i) + i''(+ 7 -lOi), - 19-12i= -2i ( 7-10i) + i2(- 1 - 2i), 7-10z = ( 3 + 5i)(- 1- 2^■) + ^■. Here 0i=-l, 02=4-1, 63 = 2, 0^ = 1, ^5 = !; so that ,0 = 2, 2^0 = 0, and QQ521 ( — i7m — T7r / = ^' or the proposed congruence is resoluble. Its four roots are + 87, +2646, as may be found by any of the indirect methods for the solution of Quadratic congruences. [Addition. In the algorithm given in the text, the remainders p2,Pi... are all uneven ; and the computation of the value of the symbol (— } is thus rendered independent of the formula (iii) of Art. 28. The algorithm given by Eisenstein is, however, preferable, although the rule to which it leads cannot be expressed with the same conciseness, because the continued fraction equivalent gg REPORT ON THE THEORY OF NUMBERS. [Art. 37. to — terminates more rapidly when the remainders are the least possible, and Pi not necessarily uneven.] 37. Cubic Residues. The Theory of Cubic Residues is less complex than that of Biquadratic Residues, and is at the same time so similar to it, that it will not be necessary to treat it with the same detail. If p be a real prime of the form 3?i + 1, and if 1, /, /- denote the roots of the congruence x^-l = Q, mod. p, the p-\ residues i'l, ^, ... ^p_i oi p divide themselves into three classes according as ^i<''"'>=l, or =/, or =/*, mod^; the first class comprising the cubic residues, the two other classes comprising the cubic non-residues. Now it can be proved that every prime number of the form 3n-|-l may be represented by the quadratic form A^ — AB-\-B^', i.e. it may be regarded as the product of two conjugate complex numbers of the forms A+Bp, A+Bp^, where p and p^ are the two imaginary cube roots of unity ; just as the theory of biquadratic residues involves the consideration of the quadratic form A^ + B'^, and of complex numbers of the tjrpe A+Bi. The real integer A^ — AB + B'^ is the norm of the complex numbers A+Bp and A + Bp^, and expresses the number of terms in a complete system of residues for either of those modules. The theory of these complex numbers has not been treated of in detail by any writer (see Eisenstein, Crelle, vol. xxvii. p. 290) ; but the methods of Gauss or Dirichlet are as applicable to them as to complex numbers involving i*. Thus it will be found that every fraction of the form -^ — j^- can be developed in a finite continued fraction, having for its quotients complex integers ; that Euclid's process for finding the greatest common divisor is applicable in this case also, and that the same arithmetical consequences may be deduced from it as in the case of ordinary integers. The prime numbers to be considered in this theory are — (1) Real primes, as 2, 5, 11, 17, &c. of the form 3n + 2. (2) Imaginary primes of the form A + Bp, having for their norms real primes of the form 3 n + 1. (3) The primes i-p, 1 -/)^ having 3 for their norm. The units are ± 1, + p, and + /s^ V A + Bp be any complex number not divisible by 1—p, it may be seen * {There is a note by Gauss on this subject in vol. ii. of his Works.} Art. 37.] REPORT ON THE THEORY OF NUMBERS. gft that of the three pairs of numbers, ±(A + Bp), ±p{A+ Bp), ±p'^{A-\- Bp), there is always one, and one only, which, when reduced to the form a + bp, satisfies the congruences a=±l, 6 = 0, mod 3. Such a number is called a primary number. The product of two primary numbers, taken {positively or} negatively, is itself primary. , If a be any prime of this theory, and Jc any number not divisible by a. Format's Theorem is here represented by the congruence k^''~^ = l, mod a. Denoting by (-) that power p' of p which satisfies the congruence ^*(A^»-i)=p», the law of cubic reciprocity is contained in the formula a and j8 denoting any two lyrimary complex primes. The demonstration of this theorem follows quite naturally from the formulae cited in Art. 30. Applying them to this particular case, we have, if p denote a real prime of the form 3 w + 1, (i) F{p).F{p^)=p, (ii) [F{p)Y=p^{p), (iii) ^{p).^{p'')=p, (iv) x|.(7i(»'-'))=0, modp ;i from which we may infer that 7*'''~i'=jo, mod -^{p). (Compare Art. 29.) In the equation (iii), -^{p) and •v|'(/9^) are primary; for from the equation \F{p)f=p^{p), it appears that ■^{p)= —\, mod 3. The congruence -yi'i'-i' =p, mod -^{p), implies that ( ?. . j =|o', whence if 7' = ^, modj), k=p—l , h ^2 F(p^)= 2 Qa^; where pi = -^{p). By these formulae the several cases of the theorem of reciprocity may be proved, as follows* : — * Eisenstein in Crelle's Journal, vol. xxvii. p. 289. But in this, as in many of his earlier researches, Eisenstein had been anticipated more than ten years by Jacobi. K 90 KEPORT ON THE THEORY OF NUMBERS. [Art. 37. First, let 5 be a prime of the form 3n + 2. Then [i^O*)]' = 'T V^) V, mod 2, or [i^(/')?^^=(|-)i>,mod?. But also [^0>)]'^' = i?i<«+''j3,*<«*^> ; so that pi(9-2)pi*(«+i) = (-2.) , mod q ; or raising each side of this congruence to the power q—1, Secondly, let ^ be a real prime of the form 3«, + 1 ; we find \_F{p)Y = {^)'F{p),m<Aq, or i^0,)«- = (^^ , mod g ; and also [i^((t))]«-i=pi<«-i>_pii<9-". Hence (— ) ( — j = ( — ) , where q, is either of the complex factors of q ; or, observing that (— ) = (— ) , and (— ) = (— ) , we may write (P£\ (Pl\ = (ML) (Ml) . It is clear from this, that if we denote the four symbols (P}\ (Pl\ (Pi) (Pi) ''9'i 3 ^5'2's ^S'l'^s ^22 '3 ^y cti, 61, 62. «2 respectively, and the reciprocal symbols by a/, 6/, 6/, ax', we have the equations Oij 61 = Oa' 61', Oj &2 = ttz' 6/, 61 6u = 6/ 62' = 1 ' which imply that Oa = a^', h^ = 62', &c., or, since a, a', b, V, ... are cubic roots of ^21^8 ^Pl'3 Art. 38.] REPORT ON THE THEORY OF NUMBERS. 91' If pi and qi be conjugate primes, the preceding proof fails ; but it is easily seen that in this case also (-) = (-) Lastly, '\£ p and q are both of the form 3» + 2, it follows from the definition of the symbols, and from Fermat's Theorem, that (-) = (-) • \q/s \p/s The complementary theorems* relating to the unit p and the prime 1 — p (which are not included in the preceding investigation) are ^ Pi ' 3 where p^ is a primary prime, and a and j8 are defined by the equality jPi = 2a-l + 3/3jo. Eisenstein has observed f that a demonstration of the law of cubic reci- procity, precisely similar to that analysed in Art. 33 of this Report, may be y'^ CiX —fr-i ^ *^d its inverse function, instead of the Lemniscate integral and Lemniscate function. He has not, however, entered into any details on this interesting subject (which is the more to be regretted, because there appears to be no published memoir treating specially y" dx —rjz. ^) ; although his latest proof of the Biquadratic law (see Art. 35) has been exhibited by him in such a form as to extend equally to Cubic Residues, and even to residues of the sixth power. [Addition. In the definition of a primary number, for 'a= +1,' read 'a = — 1.' But, for the purposes of the theory of cubic residues, it is simpler to consider the two numbers + (a + bp) as both alike primary (see Arts. 52 and 57).] 38. The first enunciation of the law of Cubic Reciprocity is due to Jacobi, and the demonstration of it which we have inserted in the preceding article * Eisenstein, Crelle's Journal, vol. xxviii. p. 28 (the continuation of the memoir cited in the preceding note). t In the memoir, ' Application de I'Algfebre ' &e., already referred to. N 2 92 REPORT ON THE THEORY OF NUMBERS. [Art. 38. 18 doubtless the same with that which he gave in his Konigsberg Lectures. In one of his earliest memoirs (' De residuis cubicis commentatio numerosa,' Crelle, vol. ii. p. 66), which was composed after the announcement, but before the pubhcation, of Gauss's memoirs on Biquadratic Residues, Jacobi had already arrived at two theorems relating to Cubic Residues, which involve the law of Reciprocity, and which he seems to have deduced from his formulae for the division of the circle. But, as it had not occurred to Jacobi, at the time when this memoir was written, to introduce, as modules, instead of the prime numbers themselves, the complex factors of which they are composed, the law of Cubic Reciprocity in its simplest form does not appear in the memoir. To complete the present account of the Theory of the Residues of Powers, or of Binomial congruences, we should have in the next place to review the recent investigations of M. Kummer on complex numbers, and on the reci- procity of the residues of powers of which the index is a prime. But the consideration of these investigations, as well as of the other researches be- longing to our present subject, our limits compel us to postpone to the second part of this Report. [Addition. Jacobi's two theorems cannot properly be said to involve the cubic law of reciprocity. K (— ) = 1, it will follow from those theorems that (— ) =1. But if (— ) =jO, or p'^, they do not determine whether (—) = p, or p\ It is remarkable that these theorems, ' formd genuina qu4 inventa sunt,' may be obtained by applying the criteria for the resolubility or irresolubility of cubic congruences (Art. 67) to the congruence r^ — 3\r — \M=0, modq (Art. 43), which, by virtue of M. Kiunmer's theorem (Art. 44), is resoluble or irresoluble according as j is or is not a cubic residue of X.] VI. REPOKT ON THE THEORY OF NUMBERS. Part II. [Report of the British Association for 1860, pp. 120-169.] 39. riESIDUES of the Higher Powers. Researches of Jacobi. — ^I'he prin- ciples which have sufficed for the determination of the laws of reciprocity affecting quadratic, cubic, biquadratic, and sextic residues, are found to be inadequate when we come to residues of the 5th, 7th, or higher powers. This was early observed by Jacobi, when, after his investigations of the cubic and biquadratic theorems, he turned his attention to residues of the 5th, 8th, and 12th powers*. It was evident, from a comparison of the cubic and biquadratic theories, that in the investigation of the laws of reciprocity the ordinary prime numbers of arithmetic must be replaced by certain factors of those prime num- bers composed of roots of unity ; and Jacobi, in the note just referred to, has indicated very clearly the nature of those factors in the case of the 5th, 8th, and 12th powers respectively. He ascertained that the two complex factors com- posed of 5th roots of unity into which every prime number of the form 5 n -f 1 is resoluble by virtue of Theorem IV. of art. 30 of this Report, are not prime num- bers, i.e. are each capable of decomposition into the product of two similar com- plex numbers; so that every (real) prime number of the form 5»-f-l is to be regarded as the product of four conjugate complex factors ; and these factors are precisely the complex primes which we have to consider in the theory of * See a note communicated by him to the Berlin Academy, on May 16, 1839, in the Monats- berichte for that year, or in Crelle, vol. xix. p. 314, or Liouville, vol. viii. p. 268, in wliich, however, he implies that he had not as yet obtained a definitive result ; nor does he seem at any subsequent period to have succeeded in completing this investigation. 04 REPORT ON THE THEORY OF NUMBERS. [ah. 40. quintic residues, in the place of the real primes they divide. To this we may add that primes of the forms 5n±2 continue primes in the complex theory; while those of the form 5«. — 1 resolve themselves into two complex prime factors. Thus 7 = 7; ll=(2 + a)(2 + a^)(2 + a'')(2 + a*); 13 = 13; 19 = (4 - 3(a + a*)) (4 - 3(a« + a»)) ; 29 = (5 - (a + a*)) (5 - (a* + a^)) ; 31 = (2 - a) (2 - a') (2 - a') (2 - a*), &C., where a is an imaginary 5th root of unity. Precisely similar remarks apply to the theories of residues of 8th and 12th powers, — real primes of the forms Sn + 1, 12n + l, resolving themselves into four factors composed of 8th and 12th roots of unity respectively. By considerations similar to those previously employed by him in the case of biquadratic and cubic residues, Jacob! succeeded in demonstrating (though he has not enunciated) the formulae of reciprocity affecting those powers for the particular case in which one of the two primes compared is a real number. But it would seem that he never obtained the law of reciprocity for the general case of any two complex primes ; and indeed, for a reason which will afterwards appear, it was hardly possible that he should do so, so long as he confined himself to the consideration of those complex numbers which present themselves in the theory of the division of the circle. No less unsuccessfiil were the efforts of ELsenstein to obtain the formulae relating to 8th powers, by an extension of the elliptical properties employed by him in his later proofs of the biquadratic theorem*. It does not appear that any subsequent writer has occupied himself with these special theories ; while, on the other hand, the theory of complex numbers composed with roots of unity of which the exponent is any prime, has been the subject of an important series of investi- gations by MM. Dirichlet and Kummer, and has led the latter eminent mathe- matician to the discovery and demonstration of the law of reciprocity, which holds for all powers of which the exponent is a prime number not included in a certain exceptional class. 40. Necessity for the Introduction of Ideal Primes. — The fundamental pro- position of ordinary arithmetic, that if two numbers have each of them no common divisor with a third number, their product has no common divisor with that third number, is, as we have seen, applicable to complex numbers formed See M. Kummer, ' Ueber die allgemeinen Reciprocitatsgesetze,' p. 27, in the Jklemoirs of the Berlin Academy for 1859. Art. 4 .] EEPORT ON THE THEORY OF NUMBERS. 95 with 3rd or 4th roots of unity, because it is demonstrable that Euclid's theory of the greatest common divisor is applicable in -each of those cases. With complex numbers of higher orders this is no longer the case* ; and it is accordingly found that the arithmetical consequences of Euclid's process, which are of so much importance in the simpler cases, cease to exist in the general theory. In particular, the elementary theorem, that a number can be decomposed into prime factors in one way only, ceases to exist for complex numbers composed of 23rdt or higher roots of unity — if, at least (in the case of complex as of real numbers), we understand by a prime factor, a factor which cannot itself be decomposed into simpler factors |. It appears, therefore, that in the higher complex theories, a number is not necessarily a prime number simply because it cannot be resolved into complex factors. But by the introduction of a new arithmetical conception — that of ideal prime factors — M. Kummer has shown that the analogy with the arithmetic of common numbers is completely restored. Some preliminary observations are, however, necessary to explain clearly in what this conception consists. 41. Elementary Definitions relating to Complex Numhers. — Let X be a prime number, and a a root of the equation — = ; then any expression of the form F{a) = a^ + a^a + a^a'+...-\-a^_^a>^-'', (A) in which a^, a^, a^, ... a^^2 denote real integers, is called a complex integral number. To this form every rational and integral function of a can always be . «>■ _ 1 reduced ; and it follows, from the irreducibUity of the equation = 0, that the same complex number cannot be expressed in this reduced form in two different ways. The norm of F{a) is the real integer obtained by forming the product of aU the \ — 1 values of F{a), so that N . F {a) = N . F {a^)= ... = N . F {a^-') = F {a) . F {a^) . F {a^) ... F {a'^-^ * {See Qauss, vol. ii, 'Zur Theorie der complezen Zahlen,' and Schering's Note.} t For complex nninbers composed with 5th or 7th roots of unity, the theorem still exists ; for 23 and higher primes it certainly fails; whether it exists or not for 11, 13, 17, and 19, has not been definitely stated by M. Kummer (see below, Art. 50). X ' Maxime dolendum videtur' (so said M. Kummer in 1844) 'quod heec numerorum realium virtus, nt in factores primos dissolvi possint, qui pro eodem numero semper iidem sint, non eadem est numerorum complexorum, qusB si esset, tota haec doctrina, quse magnia adhuc difficultatibus premitur, facile absolvi et ad finem perduci posset.' (See his Dissertation in Liouville's Journal, vol. xii. p. 202.) In the following year he was already able to withdraw this expression of regret. 96 REPORT ON THE THEORY OF NUMBERS. [Art. 41. The operations of addition, subtraction and multiplication present no peculiarity in the case of these complex numbers ; by the introduction of the norm, the division of one complex number by another is reduced to the case in which the divisor is a real integer. Thus f(a) f(a)F(a')F{a^)...F{a'-^) ^ F{a)~ N.F(a) and /(a) is said to be divisible by F(a) when every coeflScient in the product /(a) F(a^) -F(o») ... F(o^-i), developed and reduced to the form (A), is divisible by N. F{a). When /(a) is not divisible by F{a), it is not, in general, possible to render the norm of the remainder less than the norm of the divisor ; and it is owing to this circumstance that the common rule for finding the greatest common divisor is not generally applicable to complex numbers. If, in the expression (A), we consider the numbers Oo, a^, ..., a,^_2 as indeterminates, the norm is a certain homogeneous function of order X — 1, and of X — 1 indeterminates ; so that the inquiry whether a given real number is or is not resoluble into the product of X - 1 conjugate complex factors, is identical with the inquiry whether it is or is not capable of representation by a certain homogeneous form, which is, in fact, the resultant of the two forms and si^~^ + x^~^y + a^-^y^+ ...+y''~\ The problem is considered in the former aspect by M. Kummer, in the latter by Dirichlet. The methods of Dirichlet appear to have been of extreme generality, and are as applicable to complex numbers, composed with the powers of a root of any irreducible equation having integral coefiicients, as to the complex numbers which we have to consider here. Nevertheless, in the outline of this theory which we propose to give, we prefer to follow the course taken by M. Kummer : for Dirichlet's results have been indicated by him, for the most part, only in a very summary manner* ; nor is it in any case difficult to assign to them their proper place in M. Rummer's theory; while, on the other hand, it would, perhaps, be impossible to express adequately, in any other form than that which M. Kum- mer has adopted, the numerous and important results (including the law of • See his notes in the Monatsberichte of the Berlin Academy for 1841, Oct. 11, p. 280; 1842, April 14, p. 93; and 1846, March 30; also a Letter to M. Liouville, in Liouville's Journal, vol. v. p. 72; a note in the Comptes Rendus of the Paris Academy for 1840, vol. x. p. 286; and another in the Monatsberichte for 1847, April 16, p. 139. Art. 41.] EEPORT ON THE THEORY OF NUMBERS. 97 reciprocity itself) contamed In the elaborate series of memoirs which he has devoted to this subject*. * The following is a list of M. Kummer's memoirs on complex numbers : — 1. De numeris complexis qui radicibus unitatis et numeris realibus constant, Breslau, 1844. This is an academical dissertation, addressed by the University of Breslau to that of Konigsberg, on the tercentenary anniversary of the latter. It has been inserted by M. Liouville in his Journal, vol. xii. p. 185. 2. Ueber die Divisoren gewisser Formen der Zahlen, welche aus der Theorie der Kreistheilung entstehen. — Crelle, vol. xxx. p. 107. 3. Zur Theorie der complexen Zahlen, in the Monatsberichte for March 1845, or in Crelle, vol. XXXV. p. 319. 4. Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Prim- factoren. — Crelle, vol. xxxv. p. 327. The date is Sept. 1846. 5. A note addressed to M. Liouville (April 28, 1847), in Liouville's Journal, vol. xii. p. 136. 6. Bestimmung der Anzahl nicht aquivalenter Klassen fiir die aus Aten Wurzeln der Einheit gebildeten complexen Zahlen, und die idealen Factoren derselben. — Crelle, vol. xl. p. 93. 7. Zwei besondere Untersuchungen fiber die Classen-Anzahl, und fiber die Einheiten der aus \ten "Wurzeln der Einheit gebildeten complexen Zahlen. — Crelle, vol. xl. p. 117. (See also the Monatsberichte of the Berlin Academy for 1847, Oct. 14, p. 305.) 8. Allgemeiner Beweis des Fermat'schen Satzes, dass die Gleichnng a:^+2/^ = aA- unlosbar ist, filr alle diejenigen Potenz-Exponenten A, welche ungerade Primzahlen sind, und in den Zahlern der ersten J (A — 3) Bemouillischen Zahlen als Factoren nicht vorkommen. — Crelle, vol. xl. p. 131. (See also the Monatsberichte for 1847, April 15, p. 132.) This and the two preceding memoirs are dated June 1849. 9. Recherches snr les Nombres Complexes. — Liouville, vol. xvi. p. 377. This memoir contains a very full resume of the whole theory, and may be read by any one acquainted with the elements of the theory of numbers. 10. A note in the Monatsberichte of the Berlin Academy for May 27, 1850, p. 154, which con- tains the first enunciation of the law of reciprocity. 11. Ueber die Erganzungssatzc zu den allgemeinen Reciprocitatsgesetzen. — Crelle, vol. xliv. p. 93 (Nov. 30, 1851), and vol. Ivi. p. 270 (Dec. 1858). 12. A note on the irregularity of determinants, in the Berlin Monatsberichte for 1853, March 14, p. 194. 13. Ueber eine besondere Art aus complexen Einheiten gebildeter Ausdrficke. — Crelle, vol. I. p. 212 (Aug. 31, 1854). 14. Ueber die den Gaussischen Perioden der Kreistheilung entsprechenden Congruenzwurzeln. — Crelle, vol. liii. p. 142 (June 5, 1856). 15. Einige Satze iiber die aus den Wurzeln der Gleichung a^ = 1 gebildeten complexen Zahlen fttr den Fall dass die Klassenanzahl durch A theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes. — Memoirs of the Berlin Academy for 1857, p. 41. An abstract of this memoir will be found in the Monatsberichte for 1857, May 4, p. 275. 16. Theorie der idealen Primfactoren der complexen Zahlen, welche aus den Wurzeln der Gleichung O 98 REPORT ON THE THEORY OF NUMBERS. [.\rt. 42. 42. Complex Units. A complex unit is a complex number of which the norm is unity. If X = 3, there is only a finite number [six] of units included in the formula + a*. But for all higher values of X, the number of units is infinite. Nevertheless it is always possible to assign a system of m - 1 units (putting, for brevity, ^ (X — 1) = m) such that all units are included in the formula ±a''ul^u'^ ...ul'Ll^; in which Mj, «2, Wj, ... tt^_i are the assigned units, and ifc, Tii, Tij, ...n^_i are real (positive or negative) integral numbers. A system of units, capable of thus representing all units whatsoever, is called a funda- mental system. The existence, for every value of X, of fundamental systems of AX — 1 units may' be established by means of a general proposition due to Dirichlet and relating to any irreducible equation having unity for its first coefficient, and all its coefficients integral. If, in such an equation, H be the number of real, and 2 / of imaginary roots, there always exist systems of i2-|-7— 1 fundamental units, by means of which all other units can be expressed ; or, in other words, the indeterminate equation ' Norm = 1 ' is always resoluble in an infinite number of ways, and aU its solutions can be expressed by means of ^ + /— 1 fundamental solutions *. The demonstration of the actual existence, in every case, of these »" = 1 gebildet sind, wenn n eine zusammengesetzte Zahl ist. — Memoirs of the Berlin Academy for 1856, p. 1. 1 7. Ueber die allgemeinen Reciprocitatsgesetze unter den Resten und Nicht-resten der Potenzen, deren Grad eine Primzahl ist. — Memoirs of the Berlin Academy for 1859, p. 20. It was read on Feb. 18, 1858, and May 5, 1859. An abstract will be found in the Monatsberichte of the former year. A memoir by M. Kronecker (De unitatibus complexis, Berlin, 1845 ; it is his inaugural dis- sertation on taking his doctorate) connects itself naturally with the earlier memoirs of the preceding series. * To enunciate Dirichlet's theorem with precision, let f[x) = be the proposed equation; let a,, a,, ... a„ be its roots, and >/'(o,), t/'(aj), ... ■</'(a„) a system of n conjugate units. If the analytical modulus of every one of the quantities ■«/'(a,), ^{a^, ... V'(<*n) ^ unity, the system of units is an isolated or singular system. The number of singular systems (if any such exist) is always finite, whence it is easy to infer that the units they comprise are simply roots of unity. For if \// (o) be a singular unit, its powers are evidently also singular units, and therefore cannot be all different from one another; i.e. ^{a) is a root of unity. If /(a;) be of an uneven order, there are no singular units; if/(x) be of an even order, —1 is a singular unit; and if /(a;) = have any real roots, it is the only singular unit ; whereas if all the roots of / (a;) = be imaginary, other singular units may in special cases exist. Thus the equation = has 2(A.— 1) singular units included in the formula ± a*. Admitting this definition of singular units, we may enunciate Dirichlet's theorem as follows:— A system of h units [h = I+R—-[J, «, (a), e^{a), ... e^{a), composed with any root a, can always be assigned such that every unit composed with the same root can be represented (and in one way only) by the formula o) . «,»> (a) . «,«H. (a) . e,«i (a) . . . . «j"» (a). p Art. 42.] REPORT ON THE THEORY OF NUMBERS. 99 systems of fundamental units (a theorem which is, as Jacobi has said*, 'un des plus importants, mais aussi un des plus epineux de la science des nombres') is of essential importance in the theory of complex numbers, and has the same relation to that theory which the solution of the Pellian equation x^ — Dif' = 1 has to the theory of quadratic forms of determinant D. It may be observed, however, that in the case which we have to consider here, that of the equation a*^ — 1 . ,- = 0, the existence of fundamental systems of m — 1 units has been demon- a — 1 strated independently of Dirichlet's general theory by MM. Kronecker and Kummer f. If X = 5, a + a~^ is the only fundamental unit; so that every unit is in- cluded in the formula + a*(a + a~i)". If \ = 7, the complex units are included in the formula + a'^(ai + a-i)"i(a2 + a-2)»3. But for higher primes the actual calculation of a system of fundamental units involves great labour ; and a method practically available for the purpose has not yet been given. It is remarkable that every unit can be rendered real {i.e. a fiinction of the binary sums or periods a^ + a~^, &c.) by multiplying it by a properly assumed power of a. We shall therefore suppose, in what follows, that the units of which we speak have been thus reduced to a real form. For all values of X greater than 5, the number of systems of fundamental units is infinite. For if Ui, u^, ...m^_i still represent a system of fundamental units, it is evident that the system JS^i, ^2> ••• -^m-h defined by the equations E, =<•" .<^> <ir^, 1 E. =«f" ■<■'' <ir'. [ (A) is also a fundamental system, if the indices (1, 1), &c. be integral numbers, and if the determinant 2 + (1, 1) (2, 2) ... (/x — 1, yu — 1) be equal to unity. And where n^, «j , . . . ti^ are positive or negative integral numbers and to is unity, or some one of the singular units composed with a. The principles on which the demonstration of this theorem depends are very briefly indicated in the notes presented by Dirichlet to the Berlin Academy in 1841, 1842, and 1846. {See also Liouville, vol. v. p. 72.} * Crelle's Journal, vol. xl. p. 312. t See Kronecker, De unitatibus complexis, pars altera; and Kummer, in Liouville's Journal, vol. xvi. p. 323. O 2 100 REPORT ON TEffi THEORY OF NUMBERS. [Art. 43. conversely, every system of fundamental units will be represented by the equa- tions (A), if in them we assign to the indices (1, 1), (2, 2), &c. all systems of integral values in succession consistent with the condition 2±(1,1)(2,2)(3,3)...(m-1,ax-1)=+1; 80 that a single system of fundamental units represents to us all possible systems. We shall also have occasion to allude to independent systems of units. A system of m — 1 units, «,, Mj, ... w^_i, is said to be independent when it is im- possible to satisfy the equation whatever integral values are assigned to the indices n^, n2, ^3, ...«^_i. The equations (A) wUl represent all possible systems of independent units, if we suppose that in them the indices (1, 1), (2, 2), (3, 3) ... receive all positive and negative integral values, subject only to the condition that the determinant A = 2 + (1,1) (2, 2) ... (n — l, fj^ — l) does not vanish. Every system of funda- mental units is also independent ; but not conversely. Every unit can be represented as a product of the powers of the units of an independent system ; but if the system be not also fundamental, the indices of the powers are not in general integral, but are fractions having denominators which divide A. Lastly, if c^(a), 62(0), ... c^_j(a) be a system of independent units, the log- arithmic determinant X..Ci(a) , L.C2{a) , ,i.C^-i(a) Z.Ci(aT) ,L.C^{al) , ,L.C^_,(ay) in which y denotes a primitive root of X, is different from zero ; and conversely, if the determinant be different from zero, the system of units is independent. For all systems of fundamental units, the absolute value of the logarithmic determinant is the same; for any other independent system, its value is A times that least value. The quantities denoted by the symbols L.Ci(a), L . C2 (a), &c. are the arithmetical logarithms of the real units Cj (a), &c. taken positively. 43. Gauss's Equations of the Periods. — In Gauss's theory of the division of the circle, it is shown that if X be a prime number, and if e/ = X — 1, the e periods of/ roots each, that is the quantities Vo, Vu V2, ■■■ Ve-i, defined by the equations I Art. 43.] REPORT ON THE THEORY OF NUMBERS. 101 jjo =a^ +01^ + aT + +ay , tji =aT +aT +af + + a^ , (7 still denoting a primitive root of X), are the roots of an irreducible equation of order e having integral coefficients, which we shall symbolize by (see Disq. Arith., Art. 346). This equation is of the kind called Abelian ; that is to say, each of the e periods is a rational function of any other, in such a manner that we may establish the equations "Jl = (lo), V2 = 4^ ('?l), V3 = <P {12), ■■•,Vo = 'P{'Je-l)', where it is to be observed that the coefficients of the function cf) are not in general integral. The detennination of the coefficients of the equation F {y) = may be effected, for any given prime X, and any given divisor e of X — 1, by methods which, however tedious, present no theoretical difficulty. Every rational and integral function of the periods can be reduced to the form <Xo I0 + tti ri-i + a2 I2+ ...+ ««_! le^i. If we combine the equation with the e-2 equations, by which »?^, >;', ... t,''^ are expressed in that linear form, we may eliminate »?2, »?3> ••• »?«-i, and shall thus obtain an equation of order e, satisfied by »;o, i.e. the equation of the periods, or F{y) = 0. This is the method proposed by Gauss (Disq. Arith., Art. 346) ; M. Kummer, instead, forms the system of equations il = «o/+( , 0)^o + ( , l)i, + { , 2);;,+ ...+( , e-l)r,,_„ '/o'/i = «i/+( 1 , 0),o + ( 1 , l)'?i + ( 1 , 2),?2+...+( 1 , e-l)ve-i, ^ori2 = «2/+( 2 , 0),o + ( 2 , l),i + ( 2 , 2)r,,+ ... + { 2 , e-l)«7,_a, no le-i = «e-i/+ (e - 1, 0) ;7o + (e - 1, 1) »;i + (e - 1, 2) i;2 + . . . + (e - 1, e - 1) >7e-i, and eliminates ii,*i2, ■•• Ve-i from them. The symbol (k, h) represents the number of solutions of the congruence 7«i' + * = 1 + y****, mod X, x and y denoting any two terms of a complete system of residues for the modulus /: % is zero for aJl values of k, excepting that iio — l if /be even, and ni^ = \ if / be uneven*. * Liouville's Journal, vol. xvi. p. 404. 102 KEPORT ON THE THEORY OF NUMBERS. [Art. 43. The systems of equations corresponding to the particular cases e = 3, e = 4, have been given by Gauss, who has succeeded in expressing the values of the coeffi- cients {k, h) in each of those cases by means of numbers depending on the representation of \ by certain simple quadratic forms ; and has employed these expressions to demonstrate the criterion already mentioned in this Report for the biquadratic character of the number 2 *. A third method has been given by M. Libri t •" he establishes the formula in which Njt represents the number of solutions of the congruence l + x[ + x\+ ...+xl = 0, mod \ J. If S,, 8-2, S3 ... denote the sums of the powers of the roots of the equation F{y) = 0, this formula may be written thus, — \N, = \'' + S, + keS, + ^^^^e'S3+...+e/'S,^„ or, solving for 5,, Sj, ..., From this equation, when the values of iVj, N2, &c., have been determined. Si, Si, ... may be calculated, and thence by known methods the values of the coefficients of the equation F (y) = 0. Lastly, M. Lebesgue has shown that, if we denote by er^ the number of ways in which numbers divisible by X can be formed by adding together k terms of the series 7", y^, ..., 7^"*, subject to the condition that no two powers of y be added the indices of which are congruous for the modulus e, the function {\ — l)F(y) assumes the form But the practical application of any of these methods is very laborious when * Disq. Arith., Art. 358, and Theor. Res. Biq., Arts. 14-22. t See the memoir ' Sur la Th6orie des Nombres,' in his ' M^moires de Math^matique et de Physique,' pp. 121, 122. The notation of the memoir has been altered in the text. See also M. Le- besgue, in Liouville's Journal, vol. ii. p. 287, and vol. iii. p. 113. X In this congruence a;, , x.^,...x^ are k terms (the same or different) of a complete system of residues for the modulus X ; and in counting the number of solutions, two solutions are to be con- sidered as different in which the same places are not occupied by the same numbers. A simpler formula for 5j^, may be obtained by considering «, , a;, , . . . scj to represent terms of a system of residues prime to X, and denoting by e* y^ the number of solutions of M. Libri's congruence on this hypothesis. We thus find 5^+, = Xy— /* (Liouville, vol. iii. p. 116). § Liouville, vol. iii. p. 119. I Art. 44] BEPORT ON THE THEORY OF NUMBERS. 103 X is a large number, chiefly on account of the determinations which they all require of the numbers of solutions of which certain congruences are susceptible. For e = 2 the equation is y'+y + i{l-{-iy\}=0, or, putting ?' = 2y + l, r^ — { — iyx = 0. The cubic and biquadratic equations coiresponding to the cases e = 3 and e = 4 are also known from Gauss's investiga- tions. The results assume the simplest forms if we put r = ey + l. We then have (1) e = 3, 4X = ilf2 + 27iV^2, iW=l,mod3; r^-3Xr-\M=0. (2) e = 4; \ = A' + B'; ^=l,mod4; e = (-l)/ [r2 + (l-2e)Xp-4X(r-^)2 = 0*. Though these determinations are not required in M. Rummer's theory, we have nevertheless given them here, in order to facilitate arithmetical verifications of his results. The forms of the period-equations for the cases e = 8 and e = 12 can (it may be added) be elicited from the results given by Jacobi in his note on the division of the circle (Crelle, vol. xxx. pp. 167, 168). 44. The Period- Equations considered as Congruences. — An arithmetical property of the equation F{y) = 0, which renders it of fundamental importance in the theory of complex numbers, is expressed in the following theorem : 'If 2 be a prime number satisfying the congruence g'-'"=l, mod X, the congruence F(y) = 0, mod q, is completely resoluble, i.e. it is possible to establish an indeterminate congruence of the form ^(y) = iy- ^0) {y-Ui)...(y- u,_-^, mod q, tto, «i, ... M^_i denoting integral numbers, congruous or incongruous, mod jf.' * M. Lebesgue, Comptes RenduB, vol. li. p. 9. Gauss has not exhibited this last equation in its explicit form. See Theor. Res. Biq. I. c. t This theorem was first given by Schoenemann (Crelle, vol. xix. p. 306) ; his demonstration, however, supposes that q ^ e, — a limitation to which the theorem itself is not subject. The following proof is, with a slight modification, that given by M. Kummer (Crelle, vol. xxx. p. 107, or Liouville, vol. xvi. p. 408). {See also Liouville (II), vol. v. p. 369.} From the indeterminate congruence of Lagrange (see Art. 10 of this Report), x{x—l){x—2) {x—q+1) = x''—x, modg-, it follows that (zz-it) Cy-^/jt- 1) (y-it-s) . . • (2/-'?i-2+ 1) = (2/-'?t)''-(y-')i) = 2/'-V-(y-'J*) = y'-y. mody, observing that jj/ = iJi+ind ,, and that, if Ind q be divisible by e (or, which is the same thing, if q 104 REPORT ON THE THEORY OF NUMBERS. [Art. 44. a^ — 1 A particular case of this theorem, relating to the equation _ = (which may of course be regarded as the equation of the X — 1 periods, consisting each of a single root), is due to Euler, and is included in his theory of the Eesidues of Powers ; for it follows from that theory (see Art. 12 of this Report), that the x^ — 1 binomial congruence ar^ - 1 = (and therefore also the congruence = 0, mod q) is completely resoluble for every prime of the form wX + 1. A remarkable relation subsists between the periods %, tji, ... i/^.j of the equation F{i/) = 0, and the roots «o, u^, u^, ...,We_i of the congruence F{y) = 0, mod q. This relation is expressed in the following theorem : — ' Every equation which subsists between any two functions of the periods, will subsist as a congruence for the modulus q when we substitute for the periods the roots of the congruence F{y) = taken in a certain order.' It is immaterial which root of the congruence we take to correspond to any given root of the equation. But when this correspondence has once been esta- blished in a single case, we must attend to the sequence which exists among the roots of the congruence corresponding to the sequence of the periods. When «o, 1^, .••,««_! are all incongruous, their order of sequence is determined by the congruences Ui=(p{uo), U2=(p{ui), , •Wo=0K-i), mod q, which correspond to the equations and which are always significant, although the coefficients of ^ are fractional, because it may be proved that their denominators are prime to the modulus q. When Uq, «,, ..., We_i are not all incongruous [an exceptional case which implies that q divides the discriminant of F{y)'\, a precisely similar relation subsists, though it cannot be fixed in the same manner, and though the number of incongruous solutions of the congruence is not equal to the number of the satisfy the congruence ?•'= 1, mod X), rit+lni t = Vf Multiplying together the e congruences obtained by giving to k the e values of which it is susceptible in the formula (y-*?*) {y-Vt- 1) (2/-'Ji-2) . . . (y-n^-q+ l) = yi-y, mod q, we find F{y) F{y- 1) F(y-2) ... F{y-q+l) = {yO-y)', mod q; whence, by a principle to which we shall have occasion to refer subsequently (see Art. 69), it appears that •^(y) is congruous for the modulus g' to a product of the form (y-Wo)(y-Mi)-"(y-w._,). Art. 45.] REPORT ON THE THEORY OP ISTUMBERS. 105 periods. (See a paper by M. Kummer in Crelle's Journal, vol. liii. p. 142, in which he has established this fundamental proposition on a satisfactory basis*.) 45. Conditions for the Divisibility of the Norm of a Complex Number by a Reed Prime f. Instead of the complex number /(a) = «(, + ^1 a + aa a2 + . . . + a;^_2 a^-2, let us now, for a moment, consider the complex number ■^('7o) ==Co«7o +Ci»?i + C2>72+--.+Ce_i'7e_i, which, with its conjugates "^(Vi) ^Coli +CiJ72 + C2»73+...+Ce_ii7o, ^('72) =Cotl2 +Ci«73 + C2>74+---+C«_l'7l. ■^ {Ve-l)=Co '7e_i + Ci V0 + C2 '7i+ ... +C,_i 17^_2, is a function of the periods only, and is therefore a specialized form of the general complex number f{a) ; and let q still denote a real prime, satisfying the congruence q^=l, mod A. By means of the relation subsisting between the equation-roots >7o, i?!, ..., »/e-i> ^-^id the congruence-roots «o, «i, ••., «e-i> ^- Kum- mer has demonstrated the two following theorems : — (i.) ' The necessary and sufficient condition that \p- (»?) should be divisible by 2 {i.e. that the coefficients Cq, Cj, ..., Cj_i should be all separately divisible by 2) is that the e congruences ■v|'(tto) =CqUq +CiWi4-C2W2+...+Ce_i«e-i = 0, modg, ^(wi) =CoMi +CiM2 + C:jtt3-|-...-|-c'e_i«o =0, modj. ■^ (We-i) = Co We-1 + Ci t6o + C2 Ml -I- . . . + Ce_i «e-2 = 0, mod q, should be simultaneously satisfied.' (ii.) ' The necessary and sufficient condition that the norm of -^ (i?), taken with respect to the periods, i.e. the number 4"(''o)-'^('?i) ••• '*|' (»7e-i)) should be divisible by q, is that (me of the e congruences -f (tto) = 0, >!' (mi) = 0, . . . x|. (tt,_i) = 0, mod q, should be satisfied.' * {If 5" does not divide the discriminant, it is true, conversely, that if F (y) = for any value of y, mod q, q-^= 1, mod X. For we readily find {vjMq—vY =■ 0, mod q: that is q divides the discriminant, which is contrary to the hypothesis. M. Kummer shows that if q divides F{y), and y^ is not = 1, mod 2?, then g' is a residue of a power having with e some common divisor other than unity; therefore if e is a prime, 5 is a residue of an e-th power.} t The outline of the theory of complex numbers contained in this and the subsequent Articles is chiefly derived from M. Rummer's m^moire in Liouville, vol. xvi. p. 411. P 106 REPORT ON THE THEORY OF NUMBERS. [Art. 45. These results may be extended to any complex number /(a), by first reducing it to the form /(«) = 4'o (-/o) + a r|., (,o) + a^ ^2 (-/o) + • • • + a^' ' ^/- 1 ('7o). This is always possible ; for, since the / roots which compose any one period, e.g. no, are the roots of an equation x («) = ^ of order f, the coefficients of which are complex integers involving the periods only* we may simply divide /(a) by X (a), and the remainder will give us the expression of /(a) in the required form. Further, let j now denote a prime appertaining to the exponent f (not merely satisfying the congruence 2-^ = 1, mod X, but also satisfying no congruence of lower index and of the same form). The two preceding theorems are then replaced by the two following, which are analogous to them, and include them. (i.) ' The necessary and sufficient condition that /(a) should be divisible by 2, is that the congruences x/^o (%) = 0, xf'i (ttfc) = 0, . . . , v|.y:_i (wj) = 0, mod q, should be simultaneously satisfied for every value of k.' (ii.) ' And the condition that the norm of /(a) should be divisible by q, is that the same congruences should be satisfied for some one value of ^.' When the congruences ■v/^o (%) = 0, -^i (%) =0, ... xf^.i (ttfc)= 0, mod q, are simultaneously satisfied, / (a) is said to be congruoits to zero {mod q), for the substitution % = %• These / congruences may be replaced by a single congru- ence in either of two different ways. Thus, if we denote by F (rj^) the complex number involving the periods only, which we obtain by multiplying together the / complex numbers it may be proved that the single congruence F{u^)=0, mod j, is precisely equivalent to the / congruences x|.o K) = 0, x|.i (ttj) = 0, ...,^^_, («t) = 0. Or, again, if we denote by Sk (i/q) a complex number congruous to zero for every one of the substitutions io = Ui, »7o = W2> ••• '?o = We-i> b"t not congruous to zero for the substitution »7o = Mo (such complex numbers, involving the periods only, can in every case be assigned) t, it is readily seen that the same / congruences are comprehended in the single formula *k-t)/{«) = 0, modj. * Disq. Arith., Art. 348. t Crelle, vol. liii. p. 145. The number 'I' {rf) of this memoir poBseEses the property in question. Art. 46.] REPORT ON THE THEORY OF NUMBERS. 107 The utility of this latter mode of expressing the f congruences will appear in the sequel ; the formula F (u^ = 0, mod q, is of importance, because it supplies an immediate demonstration of the important proposition, that if a product of two factors be congruous to zero for the substitution r/^ = Mj, one or other of the factors must be congruous to zero for that substitution. 46. Definition of Ideal Prime Factors. — To develope the consequences of the preceding theorems, let us consider a prime number q appertaining to the exponent y"; and let us first suppose that it is capable of being expressed as the norm (taken with respect to the periods) of a complex number -v//- (j/q), which contains the periods of/ terms only ; so that 2 = ^('7o)-^('?i) ..•>f'(';«-i). If the substitution o{ u^ m. ^ render xf/- (uq) = 0, mod q, we may distinguish the e factors of q by means of the substitutions which respectively render them congruous to zero ; so that, for example, -^ {*ie-k) is the factor appertaining to the substitution % = 11^.. "We thus obtain the theorem that if /(«) be congruous to zero, mod q, for any substitution 10 = u^, f{a) is divisible by the factor of £ appertaining to that substitution. For if '^ (i/o) be that factor of q, /(g) f {a) ^{l,)^ {,,)... ^{',e-^) . ^U) q but f{a) \|/- ()7i) ^ (172) ... "v//- (i/e-i) is congruous to zero, mod q, for every one of the substitutions »7o = 'f'o> '7o = Wi> ••• "/o = ■"«-! 5 i^ is consequently divisible by q; i.e. f{a) is divisible by ^f' (i/o). A useful particular case of this theorem is that Uk-nk= 0, mod -^ (»;o), if ^ {u^ = 0, mod q. Again, it may be shown that these complex factors of q are primes in the most proper sense of the word: i.e., first, that they are incapable of resolution into any two complex factors, unless one of those factors be a complex unit ; and secondly, that if any one of them divide the product of two factors, it necessarily divides one or other of the two factors separately. That -^ (j/q) possesses the first property is evident, because its norm is a real prime, and that it possesses the second is a consequence of the last theorem of Art. 45. For if -^ (»;o) divide /i (a) x/ (a), either / (a) or / (a), by virtue of that theorem, is congruous to zero (mod q) for the substitution riQ = %', that is to say, either fi (a) or fz (a) is divisible by y^ (>;o). Now, if every prime q which appertains to the exponent / were actually capable of resolution into e complex factors composed of the e periods of/ roots, these factors would represent to us all the true primes to be considered in the p 2 108 REPORT ON THE THEORY OF NUMBERS. [Art. 47. theory of the residues of X-th powers. And for values of X inferior to 11, perhaps to 23, this is, in fact, the case. But for higher values of X, the real primes appertaining to the exponent f divide themselves into two different groups, according as they are or are not susceptible of resolution into e conjugate factors. Let, then, j represent any prime appertaining to the exponent ^^ whether sus- ceptible or not of this resolution, and let y(a) still denote a complex number which is rendered congruous to zero by the substitution j7o = Wo; ./*(«) is said by M. Kummer to contain the ideal factor of q appertaining to the substitution ijo = Uq. This definition is admissible, because it is verified, as we have just seen, when q is actually resoluble into e conjugate factors ; and its introduction is justified, as M. Kummer observes, by its utility. To obtain a definition of the multipUcity of an ideal factor, we may employ a complex number ^ (17) possessing the property indicated in the last article. If of the two congruences [^U)]" /(«) = 0,mod2», [^(,o)]'' + Y(a) = 0. mod2» + S the former be satisfied, and the latter not, f{a) is said to contain n times pre- cisely the ideal factor of q which appertains to the substitution 170 = Wq. 47. Elementary Theorems relating to Ideal Factors. — The following pro- positions are partly restatements (in conformity with the definitions now intro- duced) of results to which we have already referred, and partly simple corollaries from them. They will serve to show that the elementary properties of ordinary integers may now be transferred to complex numbers. (1.) A complex number is divisible by q when it contains all the ideal factors of q. If it contain all of those factors n times but not aU of them n + 1 times, it is divisible by g" but not by q" + \ (2.) The norm of a complex number is divisible by q when the complex number contains one of the ideal factors of q. If (counting multiple factors) it contain, in all, k of the ideal factors of q, the norm is divisible by ^■^, but by no higher power of q (/denoting the exponent to which q appertains). (3.) A product of two or more factors contains the same ideal divisors as its factors taken together. (4.) The necessary and sufficient condition that one complex number should be divisible by another is, that the dividend should contain all the ideal factors of the divisor at least as often as the divisor. (5.) Two complex numbers which contain the same ideal factors are iden- tical, or else differ only by a vmit factor. (6.) Every complex number contains a finite number of ideal prime factors. Alt. 47.] REPORT ON THE THEORY OF NUMBERS. 109 These ideal prime factors (as well as the multiplicity of each of them) are per- fectly determinate. The prime number \ is the only real prime excluded from the preceding considerations. Since \ = (1 — a) (1 — a^) ... (1 — a^~i), it appears that the norm of 1 — a is a real prime, and therefore 1 — a cannot be resolved into the product of two factors, unless one of them be a unit. Again, because the necessary and sufficient condition for the divisibility of a complex number by 1 — a is that the sum of the coefficients of the complex number should be congruous to zero for the modulus X, and because the sum of the coefficients of a product of complex numbers is congruous, for the modulus X, to the product of the sums of the coefficients of the factors, it appears that if the norm of a complex number is divisible by X, the complex number is itself divisible by 1 — a ; and also that, if the product of two complex numbers be divisible by 1 — a, one or other of the factors separately must be divisible by 1 — a. Hence 1 — a is a true complex prime, and is the only prime factor of X ; in fact, X=(l-a)(l-a2) ... (l-a^-i)=e(a)(l-a)^-', if e (a) denote the complex unit l-a2 l-a'' l-a'^-i • ...... — • , 1— a 1 — a 1— a The theorems which have preceded enable us to give a definition of the norm of an ideal complex number. If the ideal number contain the factor 1—am times, and if it besides contain k, k', k", . . . prime factors of the primes q, q, j", ... appertaining to the exponents _^ y, /",.. . respectively, we are to understand by its norm the positive integral number Xm2V2'fe'/'j"fc"/"...; a definition which, by virtue of the second proposition of this article, is exact in the case of an actually existing number. It will be observed that the number of actual or ideal prime factors (com- poimd of X-th roots of xmity) into which a given real prime can be decomposed, depends exclusively on the exponent to which the prime appertains for the modulus X. If the exponent is f, the nimiber of ideal factors is ^ = e. Thus, if 2 be a primitive root of X, q continues a prime in the complex theory ; if it be a primitive root of the congruence a;^<^"'^= 1, mod X, it is only resoluble into two conjugate prime factors. This dependence of the number of ideal prime 110 REPORT ON THE THEORY OF NUMBERS. [Art. 48. factors of a given prime upon the exponent to which it appertains is a remark- able instance of an intimate and simple connexion between two properties of the same prime number, which appear at first sight to have no immediate con- nexion with one another. It may be convenient to remark that the word Ideal is sometimes used so as to include, and sometimes so as to exclude, actually existent complex num- bers ; but it is not apprehended that any confusion can arise from this ambiguity, which it is not worth while to remove at the expense of introducing a new technical term. 48. Classification of Ideal Numbers. — An ideal number (using the term in its restricted sense) is incapable of being exhibited in an isolated form as a complex integer ; as far as has yet appeared, it has no quantitative existence ; and the assertion that a given complex number contains an ideal factor is only a convenient mode of expressing a certain set of congruential conditions which are satisfied by the coefiicients of the complex number. Nevertheless we may, without fear of error, represent ideal numbers by the same symbols, /(«), -^(a), (j) (a), ..., which we have employed to denote actually existing complex numbers, if we are only careful to remember that these symbols, when the numbers which they represent are ideal, admit of combination by multiplication or division, but not by addition or subtraction. Thus f{a) x /i (a), f{o)-^fi (a), [/(a)]", are significant symbols, and their interpretation is contained m what has preceded ; but we have no general interpretation of a combmation such as y(a)-l-_^(a), or f{a) —fi (a) *. This symbolic representation of ideal numbers is very con- venient, and tends to abbreviate many demonstrations. Every ideal number is a divisor of an actual number, and, indeed, of an infinite number of actual numbers. Also, if the ideal number (p (a) be a divisor of the actual number F{a), the quotient (p^ (a) = F (a) -i- <j) [a) is always ideal; for if (pi (a) were an actual number, <p (a), which is the quotient of F (a) divided by (pi (o), ought also to be an actual number. It appears, therefore, that there exists an infinite number of different ideal multipliers, which all render actual the same ideal number. It has, however, been shown by M. Kummer that a finite number of ideal multipliers are sufficient to render actual all ideal numbers whatever; so that it is possible (and that in an infinite number of difierent ♦ Theae symbob are, however, interpretable when /(a) and /, (a) belong to the same class. Thus, if (^(a) x/(a) and <f){a) X /, (a) be both actual, /(a)+/j (a) is the ideal quotient obtained by dividing ^ (a) xf{a.) + (p (a) x /j (o) by <^ (a). Art. 49.] REPOBT ON THE THEORY OF NUMBERS. Ill ways) to assign a system of ideal multipliers, such that every ideal number is rendered actual by one of them, and one only. Ideal numbers are thus dis- tributed into a certain finite number of classes, — a class comprehending those numbers which are rendered actual by the same multiplier ; and this distribution into classes is independent of the particular system of multipliers by which it is effected, inasmuch as it is found that if two ideal numbers be rendered actual by the same multiplier, every other multiplier which renders one of them actual will also render the other actual. Ideal numbers which belong to the same class are said to be equivalent ; so that two ideal numbers, which are each of them equivalent to a third, are equivalent to one another. We may regard actual numbers (which need no ideal multiplier) as forming the first or principal class in the distribution, and, consequently, as all equivalent to one another. If f(a) be equivalent to f^ (a), and (p (a) to ^i (a), f{a) x (p (a) is equivalent to /i (a) X <^i (a), — a result which is expressed by saying that ' equivalent ideal nimibers multiplied by equivalent numbers give equivalent products ; ' and the class of the product is said to be the class compounded of the classes of the factors. 49. Representation of Ideal Numbers as the roots of Actual Numbers. — An important conclusion is deducible from the theorem that the number of classes of ideal numbers is finite. Lety(a) be any ideal number ; and let us consider the series of ideal numbersy(a),/'(a)2,y(a)^, These numbers cannot all belong to different classes ; we can therefore find two different powers of /(a), for example [/(")]*" ^^^ [/(")]*"*"» which are equivalent to one another. But the equivalence of these numbers implies that [/(«)]" is equivalent to the actual number + ] ; i.e. that [/(«)]" is itself an actual number. We may therefore enunciate the theorem, 'Every ideal number, raised to a certain power, becomes an actual number.' The index of this power is the same for all ideal numbers of the same class, but may be different for different classes. By reasoning precisely similar to that employed by Euler in his 2nd proof of Fermat's Theorem *, it may be proved that the index of the first term in the series /(a), [/(a)]^ [/(a)]* ..., which is an actual number, is either equal to the whole number of classes, or to a sub- multiple of that number. This least index is said to be the exponent to which the class of ideal numbers containing f {a) appertains. It would seem that for certain values of the prime X, there exist classes of ideal numbers appertaining to the * See Art. 10 of this Export. 112 REPORT ON THE THEORY OF NUMBERS, [Art. 50. exponent H, if H denote the number of classes of ideal numbers *. Such classes (when they exist) possess a property similar to that of the primitive roots of prime numbers; i.e., by compounding such a class continually with itself we obtain all possible classes, just as by continually multiplying a primitive root by itself we obtain all residues prime to the prime of which it is a primitive root. It has, however, been ascertained by M. Kummer that these pnmitive classes do not in all cases, or even in general, exist. The theorem of this article enables us to express ideal numbers as roots of actually existing complex numbers. Thus, if g' be a prime appertaining to the exponent/ for the modulus X, and resoluble into the product of e conjugate ideal factors ^(>?o), ^(vi), <i>{ii), •••>^{ie-i)' these ideal numbers, which will not in general belong to the same class, will nevertheless appertain to the same exponent h ; so that [<^(»;o)]*j \jp{ii)'\''> ••• '^'^ ^^ ^® actual numbers. The power j* is therefore resoluble into the product of e actually existing complex factors. If we effect this resolution, and represent the factors of 2* by *(»?o), ^{ii), ••• the ideal numbers ip{rio), (p{tii), ... may be represented by the formulae 50. Tlie Number of Classes of Ideal Numbers. — The number of classes of ideal numbers was first determined by DLrichlet. He effected this determination by methods which he had previously introduced into the higher arithmetic, and which had already led him to a demonstration of the celebrated theorem, that every arithmetical progression, the terms of which are prime to their common difference, contains an infinite number of prime numbers ; and to the determina- tion of the number of non-equivalent classes of quadratic forms of a given detenninant f. Dirichlet's investigation of the problem which we are here * See on this subject M. Kummer's note ' on the Irregularity of Determinants ' in the Monats- berichte of the Berlin Academy for 1853, p. 194. M. Kummer's investigation, however, is restricted to classes containing ideal numbers /(a) such that /(a) x /(a~') is an actual number. + See his memoirs on Arithmetical Progressions, in the Transactions of the Berlin Academy for the years 1837 (p. 45) and 1841 (p. 141), or in liouville, vol. iv. p. 393, ix. p. 255. The first of these papers relates to progressions of real integers, the second to progressions of complex numbers of the form a + bi. In the memoir 'Recherches sur diverges applications de I'analyse infinit^simale 4 la Thferie des Nombres' (Crelle, vol. xis. p. 324, xxi. pp. 1 & 134), Dirichlet has applied his method to quadratic forms having real and integral coeflBcients ; and in a subsequent memoir (Crelle, vol. xxiv. p. 291) he has extended this application to quadratic forms, of which the coefficients are complex numbers containing t. See also Crelle, vol. xviii. p. 259, xxi. p. 98 (or the Monatsberichte for 1840, p. 49), xxii. p. 375 (Monatsberichte for 1841, p. 190). "We shall have occasion, in a later part of this Report, to give an abstract of the contents of this invaluable series of memoirs. Art. 50.] REPORT ON THE THEORY OF NUMBERS. 113 considering has never been published ; but that since given by M. Kummer is probably in all essential respects the same, as it reposes on an extension of the principles developed in Dirichlet's earlier memoirs. Our limits compel us to omit the details of M. Kummer's analysis ; the final result, however, is that, if H denote the number of non-equivalent classes of ideal numbers, H = D In this formula P is a quantity defined by the equations P = <^(/3).<^(;83).<^(/3«)...<^0^-^), 0(|8) = 1 +7x^ + 72 ^' + 73/3^+. .•+rA-2^'-', /3 representing a primitive root of the equation j8^~^ = 1, 7 a primitive root of the congruence y^^'^sl, mod \, and 71, 72; 73 > ••• the least positive residues of 7. 7^> 7'> ••• foi" the modulus X ; A is the logarithmic determinant (see Art. 42 of this Report) of any system of ^ — 1 fundamental units, and D the logarithmic determinant of a particular system of independent but not fundamental units, e(a), e(aT), e{ay^), ...,e{ay^~^), defined by the equation rM - A / (^ -°'^ (^ -"'') ■ °'''"^ (^ -"') ■ '^"^ Ifa-.^- «W-V a-«Hl-«-n -^ UTa -±-;-^, ifa-e ^ . so that sm — A D = L.e{a) , Z.e(aT) , L.e{ay'),...,L.e{ay''-') L.e{ay) , L.e{ai') , Z.e(a7^), ...,Z.e(a7''-^) L.e{a^^) , L.e{ay') , i:.e(aT'*), ...,Z.e(ai"') X.e(aT"'-'), L.eiay"-'), L.e{a'"'),...,L.e{ay"'-') Each of the two factors P D . jr—T — - and -7-, of which the value of H is com- (2\)'*-i A' posed, is separately an integral number. That -r is integral is a consequence of the relation which exists between the logarithmic determinant of a system of fundamental units, and that of any system of independent units ; that P is divisible by (2X)''~^ may be rendered evident from the nature of the expression P itself*. The factor -r-, taken by itself, represents the number of classes that contain ideal numbers composed with the periods of two terms a + a" ', o^ + a" ^, • • . * See the investigation in the next article. Q 114 REPORT ON THE THEORY OF NUMBERS. [Art. 51. only; or, which is the same thing, it represents the number of classes each of which contains the reciprocal /(a" >) of every ideal number /(a) comprehended in p it • ^ — r — , on the other hand, is the number of classes of those ideal nxmibers ' (2\)''-* which become actual by multiplication with their own reciprocals *. The actual calculation of the factor j- is extremely laborious, as it requires the preliminary investigation of a system of fundamental units. For the cases X = 5, X = 7, the trigonometrical units e(a), e{a»), e^ai^) ... are themselves a fundamental system, so that in these two cases D = A, and x = + 1- The computation of the first p factor T^r-r — - presents somewhat less difficulty; and M. Kummer (though not without great labour) has assigned its value for all primes inferior to 100. For the primes 3, 5, 7, 11, 13, 17, 19, that value is unity; for 23 it is 3, and then increases with extraordinary rapidity; so that for 97 it already amounts to 411322823001 = 3457 x 118982593. The asymptotic law of this increase is ex- pressed by the formula ^""L(2X)^"2i^^^J = ^' when X increases without limit f . It will be seen that the number of classes of ideal numbers for X = 3, X = 5, X = 7, is unity; i.e., for those values of X every complex prime is actual. In the absence of any determination of a system of fundamental units for X = 11, X = 13, X = 17, and X = 19, it is not possible to say whether this is or is not the case for these values also. But from and after the P . . limit X = 23, the value of the factor , . indicates that a complex number is not necessarily a complex prime because it is irresoluble into factors. 51. Criterion of the Divisibility of H by X. — The number of classes of ideal numbers, which we have symbolized by H, is not in general divisible by X ; but in certain cases it may happen that it is so. The quotient — is never divisible P . ... by X, except when the other factor . . _^ is also divisible by X. And it has been found by M. Kummer that the necessary and sufficient condition for the * See the note already cited, ' on the Irregularity of Determinants,' in the Monatsberichte for 1853, p. 195. t LiouTille, vol. xvi. p. 473. The formula is given without demonstration. Art. 51.] EEPOET ON THE THEORY OF NUMBERS. 115 . . . . P divisibility of ,_ > _^ by X is that the numerator of one of the first /w - 1 functions of Bernoulli should be divisible by X. The investigation of this singular criterion depends on a transformation of the function <^(/3) which enters into the expression of P. If we represent the product (yp - 1) <(>{^) = (yy;,., - 1) + (7 - ri)/3 + (rn - 72)^^ + ... + (77^-3 - 7^-2)^^-'. in which every coefficient is divisible by X, by X[6o + &i/3 + &2^^+...6a-2|8'-'], or Xx|.(/3) (6m denotiog the quotient Zls=i — 2^, or I^-^—i, if /represent the greatest in- X X teger contained in the fraction before which it is placed), we obtain by multlpli- "t^tion the equality (7''+l)P = X''^|.(;8).^|,(/33) ... x|,(/3^-^) ; or, since yi^ + l is divisible by X, and may be supposed not divisible by X^ *, p C denoting a coefficient prime to X. The congruence , . _^ = 0, mod X, is there- fore equivalent to the congruence >f. (;8).^|, (^3). ..^(^x-.) = o, mod X, which may, in its turn, be replaced by the following, ■f (?) . ^ (?') •••"*!' (7'"^) = 0, mod X. For, if there be an equation which, considered as a congruence for a given modulus X, is completely resoluble for that modulus, any symmetrical function of the roots of the congruence is congruous, for the modulus X, to the cor- responding function of the roots of the equation. The function which is a symmetric function of j8, ^^, ... /S^"^, the roots of the equation a;'' 4-1 =0, is therefore congruous to '^ (7) . '*|' (7^) ... '*|'(7^~*)> which is the same function of y, y^, 7*, ... 7^~^ the roots of the congruence x''-t-l = 0, mod X. * For •f'-'rl and {y + xy+l are both of them divisible by X; but only one of them can be- divisible by X', since their difference is not divisible by X^ We can therefore, without changing Vo' yii •••) yv-s) determine y in accordance with the supposition in the text. Q2 116 REPOET ON THE THEORY OF NUMBERS. [Art. 51. Hence the necessary and sufficient condition for the divisibility of % ^^ by X is that one of the n congruences included in the formula >^(72»-i) = 0,modX, w = l, 2, 3, ...,X, (a) should be satisfied. Now or, observing that 7o> 7i. Te. ••• 7x-2 ^^ the numbers 1, 2, 3, ... X — 1, taken in a certain order, and introducing the values of 6o> &i. bz, ... ^-(2«-i)^(^2n-i) = '" 2 sc'^-^l'^, mod X. X~l A This last expression may be further transformed as follows. I£ f{x) denote any x=x function of x, and F{x)= 2 f{x), we have the identical equation a= 1 7 and X being any two numbers prime to one another. To verify this equation, we may construct a system of unit points in a plane ; then the right-hand member is the sum of the values of /(a;) for all unit points in the interior of the parallelogram (0, 0), (X, 0), (X, 7), (0, 7) ; while the two terms of the left-hand member represent similar sums for the two triangles into which the parallelogram is divided by its diagonal •yx — \'y = Q. Writing then in this identity a;^""^ for x = x f{x), and employing the symbol i^2n-i i^) to represent the sum 2 x^"~^, or rather the function "=^ ... + ^ I) ^«-iji(2)n(2n-2)'^' in which B^, B^, ...,Bn are the functions of Bernoulli, and which, when x is an integral number, coincides with that sum, we find 2 X— /:^+ 2 i^.„_J/^l=(7-i)F2„-i(^-i). *=T-1 - r ^Xx- «=i ^ x=i L r But JFsn.i (X - 1) = jPa„_i (X) - X^"-^ is evidently divisible by X ; so that rXX" 7 ie=\-l a;=T-l '2 x^-i/:^-!- 2" i?;„.J/^l = 0,modX. x=l X ^,1 L 7 J Art. 52.] REPORT ON THE THEORY OF NUMBERS. 117 The congruences (a) may therefore be replaced by the congruences x=l L 7 J which may be written in the simpler form "TV2„_i(--) = 0,modX, if we observe that (\ being prime to y) the numbers /— , / — , ,.., 7-^^^^ '— 7 7 7 are congruous (mod X) to the fractions — , — , ... - , taken in a certain 7 7 7 order. But, by a curious property of the function i^2»-ij demonstrated for the first time by M. Kummer, "^v\ ( ?\ _ (-i)::M7!:^-i) The condition for the divisibility of if by X is therefore that one of the m con- gruences included in the formula 5„ (7^" — 1) = 0, mod X, should be satisfied. The last of these congruences, or ^^(7^'' — 1) = 0, is never satisfied; for it is easily proved that the denominator of B^ contains X as a factor, while 7"'-l = (7'' + l)(7''-l)> though divisible by X, is not divisible by X^. And since, iS n<m, 72" - 1 is prime to X, that factor may be omitted in the remaining m — 1 congruences ; so that the condition at which we have arrived coincides with that enunciated at the commencement of this article. We have exhibited M. Rummer's analysis of this problem with more fulness of detail than might seem warranted by the nature of this Report, not only on account of its elegance, but also because it exemplifies transformations and pro- cesses which are of frequent occurrence in arithmetical investigation *. 52. 'Exceptional' Primes. — A prime number X, which, like 37, 59, and 67 in the first hundred, divides the numerator of one of the first 2(^ — 3) functions of * In Liouville, vol. i. (New Series) p. 396, M. Kronecker has given a very simple demonstration of the congruence 2n\i/^ (y»»->) = (/"- 1) [!='» + 2»"+ ... +(X- 1)'"], mod X^ which, combined with another easily demonstrated formula, viz., ljn^2*»+...+(\-l)=" = (-l)"-'.S„X,modA«[n<fi], leads immediately to the theorem of M. Kummer. 118 REPORT ON THE THEORY OF NUMBERS. [Art. 53. Bernoulli, and which consequently divides the number of classes of ideal numbers composed with \-th roots of unity, is termed by M. Kummer an exceptional prime. Such primes have to be excluded from the enunciation of several important propositions ; and their theory presents difficulties which have not yet been overcome. Thus the following propositions are true for all primes other than the exceptional primes, but are not true for the exceptional primes. (1.) The exponent to which any class of ideal numbers appertains (see Art. 49) is prime to \. (2.) The index of the lowest power of any unit which can be expressed as a product of integral powers of the trigonometric units is prime to X. For that index is a divisor of -^ (see Art. 42). (3.) Every complex unit which is congruous to a real integer for the modulus X is a perfect X-th power. (Whether X be an exceptional prime or not, the X-th power of any complex number is congruous, for the modulus X, to a real integer, viz. to the sum of the coefficients of the complex number.) (4.) K f(a) denote any (actual) complex number prime to X ({. e. not divisible by 1 — a), a complex unit e (a) can always be assigned, such that the product F(a) = e{a)f{a) shall satisfy the two congruences F{a) F(a-^) = [F{1)]\ mod X, i^(a) = i^(l),mod(l-a)^ A complex number satisfying these two congruential conditions is called a primary complex number ; the product of two primary numbers is therefore itself primary. This definition, in the particular case X = 3, includes the primary numbers of Art, 37, taken either positively or negatively. 53. Fermat's Theorem for Complex Primes. — Let (p (a) be an actual or ideal complex prime, and let N=N.(I) (a) represent its norm. A system of N actual numbers can always be assigned such that every complex number shall be con- gruous to one and only to one of them for the modulus (p (a). These N numbers may be said therefore to form a complete system of residues for the modulus (p (a) ; and by omitting the term divisible by <j) (o), we obtain a system of iV— 1 residues prime to <{> (a). Let 5 be a prime appertaining to the exponent yj so that N=q'^, and let ^ (a) or ^1 (»;o) be the prime factor of q which appertains to the substitution >jo = Mo ; the formula ao+aia + a2a^+...+a/_ia^-^ (A) Alt. 53.] EEPORT ON THE THEORY OF KUMBERS. 119 will represent a complete system of residues for the modulus (pi (j^o), if we assign to the coefficients Oo, Oj, Og, ... the values 0, 1,2, ...,q-l in succession. For if be any complex number, /(a) is congruous for the modulus ^i (>?o) to >fro (Uo) + axf^i (tto) + . . . + a/-i x|.^_i (^o), because Uo — Vo = 0, mod (p^ {%) : that is, /(a) is congruous to one of the complex nimibers included in (A) ; nor can any two numbers ao + aia + a2a^ + ...+af_ia'^-^ and ho + hia + \a^ + ... + hf_ia'f-^ included in that formula be congruous to one another ; for the congruence (tto - 60) + « («i - &i) + a2 (ttg - 62) + . . . + a-^-^ {a/_i - hf_i) = 0, mod (p^ {rio), involves, by M. Kummer's theory (see Art. 45), the coexistence of the / congruences ao — ho = 0, mod q; tti — 6i = 0, mod g; ; ay_i — &y_i = 0, mod g; i.e. the identity of the complex numbers ao + aai + a^a2+...+a-^~^af_i, and ho + ahi + a^b2+ ...+a^~'^hf_j. It is worth while to notice that, if g be a prime appertaining to the exponent 1, for the modulus X, that is if q be of the linear form mX + 1, the real numbers 0, 1, 2, 3, ..., q — 1 will represent the terms of a complete system of residues for the modulus (p (a) ; but if (p (a) be a factor of a prime appertaining to any higher exponent than unity, a complete system will contain complex as well as real integral residues. By applying the principle (see Art. 10) that a system of residues prime to the modulus, multiplied by a residue prime to the modulus, produces a system of residues prime to the modulus, we obtain the theorem, which here replaces Fermat's Theorem, that if -^ (a) be any actual number prime to (p (a), [^{a)Y-'^ = l,mod(p{a). K we combine with this theorem the principle of Lagrange (cited in Art. 11) which is valid for complex no less than for real prime modules, we may extend, mutatis mutandis, to the general complex theory the elementary propositions relating to the Residues of Powers, Primitive Roots, and Indices, which, as we have seen, exist in the case of complex primes formed with cubic or biquadratic roots of unity. In fact, these propositions are of a character of even greater generality, and may be extended, not only to complex numbers formed with roots of unity whose index is a composite number, but also to all complex 120 BEPORT ON TBE THEORY OF NUMBERS. [Art. 64. numbers formed with the roots of equations having integral coefficients, as soon as the prime factors of those complex numbers are properly defined. 54. M. Kummer's Law of Reciprocity. — We can now enunciate M. Rummer's law of reciprocity. It appears, from the last article, or it may be proved N-1 immediately by dividing the N-l residues of <p{a) into \ groups of terms, after the following scheme, (0) n, rj, , ry_i, K (1) ari, ar^, , «»'iyr_i, X (2) a^n, a^r„ , g'ry.i. (X-1) a^'^Ti, a^-^r^, , a^~^r A. and proceeding as in Art. 33 of this Report, that if -^ (a) be any actual complex iV-l number prime to (j> (a), \fr (a) *• is congruous for the modulus (p (a) to a certain power a* of a. This power of a may be denoted by the symbol ^-j4 ; so that we have the congruence [^^(a)]-^ = [i|^] = «^mod<^(a). The symbol ^, '. , which we may term the X-tic character of -^ (a) with regard to <f) (a), is evidently of the same nature as the corresponding symbols with which we have already met in the quadratic, cubic, and biquadratic theories, and admits of an extension of meaning similar to that of which they are susceptible. Availing himself of this symbol, M. Kummer has expressed his law of reciprocity by the formula .- i / \-, ^.^i \- r^m _ rywi L<^(a)JrU(a)Jx' (p (a) and •vf' (a) denoting real or ideal primes. But, to interpret this equation rightly, it is important to attend to the following observations. (1) When •v//' (a) and (p (a) are both actual numbers, the formula supposes that they are both primary prime numbers. The prime 1 —a is therefore excluded. (2) The definition that we have given of the symbol j ^ / becomes un- Art. 55.] BEPORT ON THE THEORY OF NUMBERS. 121 meaning when <p (a) is ideal, because no signification can be assigned to an ideal number which presents itself, not as a modulus or divisor, but as a residue. Let, therefore, h denote the index of the lowest power of <p (a) which is an actual number ; i.e., let h be the exponent to which the class of cf) (a) appertains ; and let [(p (a)]* represent the actually existing primary complex number which con- tains the factor (p (a) h times, but contains no other prime factor ; then the symbol jVr has by the preceding definition a perfectly definite meaning. Let then -yVr = «*^ ; we may define the value of the symbol ^ ) I by means of the equation ^ <^ (a) Y'_ r {afl _ ,,. which, if h be prime to X, always gives a determinate value o*' for f ) i . k being defined by the congruence hk = k', mod X. For the symbol . , ^ so defined, the law of reciprocity stUl subsists, subject however to the condition that [(p (a)]* is primary. It will be seen, therefore, that the exceptional primes of Art. 52 are ex- cluded from M. Rummer's law of reciprocity, for a twofold reason : — first, because if X be one of those numbers, the definition of a primary number is not in general applicable ; and secondly, because, on the same supposition, the symbol . , , may become unmeaning. 55. TTie Theorems complementary to M. Kummer's Law of Reciprocity. — The prime 1 — a, and its conjugate primes, as well as the complex units, are excluded from the law of reciprocity ; but complementary theorems by which the X-tic characters of these numbers may be determined have been given by M. Kummer. For a simple unit aJ', we have the formula With regard to X, which is the norm of 1 — a, it may be observed that if cp (a) be a prime factor of a real prime q appertaining, for the modulus X, to any exponent /different from unity, i.e. if g- be not of the linear form mX-l-1, the character of every real integer, and therefore of X, with respect to (p (a) is +1, because, if / > 1, ^-- — is divisible by g- 1- But whatever be the linear form of q, the 122 REPORT ON THE THEORY OF NUMBERS. [Art. 55. characteristic of X or x (X) (for so we shall for brevity term the index of a in the equation \ . , | = o') is determined by the congruence X(X) = ^A, modX, d^ loff (b (e") Dx. being the value (for t; = 0) of the differential coefficient ^T^ ' , if ^ (o) be an actually existent number, or of t "l^ -^ if it be ideal. To obtain the characteristics of the units, M. Kummer considers the system of independent "nit« EM^E,{a\ ,^^_,(a), defined by the formula E^ (a) = e (a) e (a7)T'-''e (a^y'*' e {at^-y''-''\ in which e (a) represents the trigonometrical unit of Art. 50, and 7 is the same primitive root of X which occurs in the expression of e (a). We have then, for X \E]t (o")] and x (^ ~ "')> '^'^ formulae X \E, (««)] = ( - 1)^ {t' - 1) ^ D.-n, mod X, and T) N—'\ Tc'^ l-i 1-^-3 x(i-a*)=-^+i^+AA-2|-5.A-.j + ... + (-i)''5.-iA^. N representing the norm of ^ (a), B^, B^, ..., -B^_i the functions of Bemoidli, and D„^ the value of the differential coefficient d-^ogH^^) / cZ"'log[0(e'')]\ ^^^^^^ dv"" V hdv*" ) These formulae do not in general hold for the exceptional prime numbers X, which divide the numerator of one of the first m — 1 functions of Bernoulli. This is evident from the occurrence in them of the coefficients Z),„, which if <^(a) be ideal, and h be divisible by X, may acquire denominators divisible by X, thus rendering the congruences nugatory. It is sufficient to have determined the characteristics of the particular system of units Ei{a), E^ia), ..., ^^_i (a), be- cause, as that system is independent, every other unit e{a) is included in the formula , („) = ^^ („)», ^^ („)™, ^^_^ („)»^_. . so that X [« (a)] may be found from the congruence fc=/i-i X [e (a)] = 2 mi X [Ek (a)] , mod X, which cannot become unmeaning, except in the case of the exceptional prunes ; Art. 56.] REPORT ON THE THEORY OF NUMBERS. 123 because If D' be the logarithmic determinant of the system of units Ei (a), E<^ (a), ..., -fiJ^.i (a), D and A retaining the meanings assigned to them in Art. 50, it may be shown that -jr is prime to X, and therefore ■x' ~ 7T ^ a" ^ ^^ prime to X ; i.e., the denominators of the fractions m^, m^, ..., m^_i are prime to X (see Art. 42). But M. Kummer has also given a formula which assigns directly the characteristic of any unit e (a) whatsoever. If Aj. denote the value of the differential coefficient , . ^ ' , for v = 0, we have air XK«)] = ^^+ "2~A,,7),_,„modX* 56. We have already observed (see Art. 39) that it is impossible to deduce a proof of the highest laws of reciprocity from the formulae which present themselves in the theory of the division of the circle. It is true (as we shall presently see) that the formulae IV. and V. of Art. 30 determine the decom- position of the real prime p (supposed to be of the form ^X + 1) into its X — 1 complex prime factors ; but it will be perceived that these complex factors occur, not isolated, but combined in a particular manner. From equation IV. of the article cited we infer that 2^ — ^ (") ^ ("~0 5 1^^ then ^f'(«) = /(«0/(«.)-/(s); aj, Og, ..., a^ being fi different roots (of which no two are reciprocals) of the a^-1 equation - = 1 ; so that f{a^,fia^, ...,f[a^ are one-half of the complex primes of which p is composed ; if e (a) be any real unit, satisfying the equation e(a) = e (a~ '), it is plain that e (ai)2 e {a^f ...e (a^)^ = 1, or >/' (a) = + e (ai)/(ai) X e {a^fia^ X ... X e (a^)/(s)- The consideration, therefore, of the number \|/' (a) cannot supply us with any determination of the X-tic character of /(oj) which will not equally apply to /(oj) X e (ai). But for all values of X greater than 3, the number of real complex units is, as we have seen, infinite ; and the character of any complex prime f{a) with respect to any other complex prime evidently changes when /(«) is mul- tipUed by a unit of which the X-tic character is not unity. The inapplicability of the formulae of Art. 30 to any general demonstration of the law of reciprocity ♦ The fommke of this article are taken from M. Rummer's second memoir on the complementary theorems (Crelle, toI. IvL p. 270). R 2 124 REPORT ON THE THEORY OF NUMBERS. [Art. 67. is thus apparent. The only equation of reciprocity that has been elicited from them is the following : — L (7, Jx L 52 Jx L q, 4 l-<^(«)-lx L<^(a)Jx L<^(a)Jx in which <^ (a) is a complex prime factor of a prime number p of the form mX + 1, and g,, q^, ...,qt are the e conjugate factors of a prime number q appertaining to the exponent / for the modulus X. This equation, which, if we adopt the generalised meaning of the sjmabol of reciprocity, may be written more briefly thus, 1^-^ I = I ^ was first obtained by Eiseiistein, who inferred it from L 9 Jx L0(a)Jx M. Rummer's investigation of the ideal prime divisors of xf^ (a) (see a note ad- dressed by Eisenstein to Jacobi, and communicated by Jacobi to the Berlin Academy, in the Monatsberichte for 1850, May 30, p. 189). In a later memoir (CreUe's Journal, vol. xxxix. p. 351), Eisenstein proposes an ingenious method — reposing, however, on an undemonstrated principle — for the discovery of the higher laws of reciprocity; but it would seem that the application of this method failed to lead him to any definite result ; and it is unquestionably to M. Rummer alone that we are indebted for the enunciation as weU as for the demonstration of the theorem. 57. M. Rummer appears to have waited until he had developed the theory of complex numbers with a certain approximation to completeness, before pro- ceeding to apply the principles he had discovered to the purpose which he had in view throughout, the investigation of the law of reciprocity. He suc- ceeded in discovering the law which we have enunciated, in the year 1847, and, after verifying it by calculated tables of some extent, he communicated it to Dirichlet and Jacobi in January 1848, and subsequently, in 1850, to the Berlin Academy, in a note which also contained the demonstration of the com- plementary theorems relating to the units, and the prime divisors of X. From the analogy of the cubic theorem, it was natural to conjecture that the law of reciprocity would assume the simple form ~ ~ ^^^ primes p^ and p^ reduced, by multiplication with proper complex units, to a form satisfying certain congruential conditions. But to determine properly these conditions, i.e. to assign the true definition of a primary complex prime, was no doubt the principal difficulty that M. Rummer had to overcome in the discovery of his theorem. K X = 3, the single congruence /(a)=/(l), mod(l-a)2, suffi- ciently characterises a primary number; and since, whatever prime be repre- I Art. 58.] EEPORT ON THE THEORY OF NUMBERS. 125 sented by X, that congruence is satisfied by one, and one only, of the numbers included in the formula a!'f{a.), it was probable that it ought to form one of the congruential conditions included in the definition of a primary complex prime. In determining the second condition, M. Kummer appears to have been guided by a method which depends on the arithmetical properties of the log- arithmic expansion of a complex number. If we develope log vrTT in ascending powers of ^^.:! and represent by Ljjty the finite number of terms which remain in this expansion after rejecting those which are congruous to zero for the modulus X, we are led, after some transformations, to the congruence where X^ (a) represents the function 2 -y-'* a", and Z)j. denotes, as in Art. 55, 8 = d'^ log: f (e'") the value (for v = 0) of the differential coefficient f^, . In this con- gruence the first coefficient alone is altered when f{a) is multiplied by a simple unit ; and only the even coefficients are altered when f{a) is multi- plied by a real unit. Now Di is rendered congruous to zero by the condition / (a) = f (I), mod (1 — ay ; and M. Kummer has shown that, by multiplying f{a) by a properly chosen real unit, D^, D^, ..., D)^_3 may be similarly made to disappear, so that we obtain a congruence which is proved to involve the second congruence of condition satisfied by a primary number, i.e. f{a)f(a~^) =f{yf) mod. X *. 58. The methods to which M. Kummer at first had recourse in order to obtain a demonstration of his theorem, consisted in extensions of the theory of the division of the circle. By such extensions he demonstrated the comple- mentary theorems, and even a particular case of the law of reciprocity itself — that in which the two complex primes compared are conjugate. But, after repeated efforts, he found himself compelled to abandon these methods, and to seek elsewhere for more fertile principles. ' I turned my attention,' he says, 'to Gauss's second demonstration of the law of quadratic reciprocity, which depends on the theory of quadratic forms. Though the method of this demon- * Crelle, vol. xliv. pp. 130-140. 126 REPORT ON THE THEORY OF NUMBERS, [aH. 59. stration had never been extended to any other than quadratic residues, yet its principles appeared to me to be characterised by such generality as led me to hope that they might be successfully applied to residues of higher powers ; and in this expectation I was not disappointed *.' M. Rummer's demonstration of the law of reciprocity was communicated to the Academy of Berlin in the year 1858, ten years after the date of his first discovery of it. An outline of the demonstration is contained in the Monatsbe- richte for that year ; and it is exhibited with great clearness and fulness of detail in a memoir published in the Berlin Transactions for 1859, which con- tains what is for the present the latest result of science on a problem which, if we date from the first enunciation of the quadratic theorem by Euler, has been studied by so many eminent geometers for nearly a century. It would, however, be impossible, without exceeding the limits within which this Report is confined, to give an account of its contents, which should be intelligible to persons not already familiar with the subject to which it refers. Taken by itself the demonstration of the theorem is, indeed, sufficiently simple ; but it is based on a long series of preliminary researches relating to the complex numbers that can be fonned with the roots of the equation w'^^D (a), in which D (a) itself denotes a complex number composed of Xth roots of unity. To those researches, and to the demonstration of the law of reciprocity founded on them, we shall again very briefly refer, when we come to speak of the corresponding investiga- tions in the theory of quadratic forms, an acquaintance with which is essential to a comprehension of the method adopted by M. Kummer in his memoir. We may add that M. Kummer has intimated that he has already obtained two other demonstrations of his law of reciprocity, which, though they also depend on the consideration of complex numbers containing w, yet do not require the same complicated preliminary considerations. 59. Complex Numbers composed of Roots of Unity, of which the Index is not a Prime. — In a special memoir (see the list in Art. 41, note. No. 16), M. Kummer has considered the theory of complex; numbers composed with a root of the equation w" = 1, in which n denotes a composite number. The primitive roots of this equation are the roots of an irreducible equation of the form ^ (.n_l)n (...», -1)... ^ V ' n n ' n(wp-i) n(w^r^s-i) ... See the Berlin Transactions for 1859, p. 29. Art. 59.] EEPORT ON THE THEORY OP NUMBERS. 127 Pi, P2, Pa, •■• denoting the different prime divisors of w*. If yp (n) be the number of numbers less than n and prime to it, F(w) is of the order -^{n), and every complex number containing w can be reduced (and that in one way only) to the form y(„) = a, + a,w + a,w'+ ...+ a^^„^_, wI'M-i, The numbers conjugate to /(«) are the -vf' (n) numbers obtained by writing in succession for w the -^ (n) primitive roots of «" = 1 ; and the norm of /(co) is the real and positive integer produced by multiplying together the ^ («) conjugates. If 5- be a prime number not dividing n, the sum in which the series of terms is to be continued until it begins to repeat itself, is termed a period. The n periods i^i, •sra, ..., ■sr^ remain unchanged if for w we write afl, ufl"^, etc. Hence, if q appertain to the exponent t for the modulus n {i.e. if q satisfy the congruence 2' = 1, mod n, but no congruence of a lower order and similar form), the number of different numbers conjugate to a given complex number containing the periods only is at most • For brevity, a complex number containing the periods only — for example, the number may be symbolised by/(wi), so that /(wi) = Co + Ci wj + C2 isTaj + . . . + C„ w„j. . If 1, Vi, r^, ... are a set of — ^ numbers prime to n and such that the quotient of no two of them (considered as a congruential fraction f) is congruous for the modulus n to any power of q, the numbers conjugate tofizi) may be represented a;"— 1 * The irreducibility of the equation — = when ti is prime was firat established by Gauss *c ~~ 1 (Difiq. Arith., Art. 341). For other and simpler demonstrations of the same theorem, see the memoirs of MM. Kronecker (Crelle, xxix. p. 280, and Liouville, 2nd series, vol, i. p. 399), Schonemann (Crelle, vol. xxxi. p. 323, vol. xxxii. p. 100, & vol. xl. p. 188), Eisenstein (Crelle, vol. xxxix. p. 166), and Serret (Liouville, vol. xv. p. 296). The principles on which these demonstrations depend suffice to a^^-l establish the irreducibility of thfi equation •' ,„_, = 0, but they fail, as M. Kronecker has observed, as*" —1 to furnish the corresponding demonstration when «, as in the text, is a product of powers of different primes. This demonstration was first given by M. Kronecker (Liouville, vol. xix. p. 177), who has been followed by M. Dedekind (Crelle, vol. liv. p. 27), and by M. Arndt (i6. Ivi. p. 178). t For the definition of a congruential fraction see Art. 14. 128 REPORT ON THE THEORY OF NUMBERS. [Art. 59. by /(w,), /(wr,), /(''r,). •••• The periods are the roots of certain irreducible equations, each of which is completely resoluble when considered as a congru- ence for the modulus 2 ; and the roots Ui, u^, ... of the congruences are connected with the roots •a^, nr^, ... of the equations, by a relation precisely similar to that enunciated in Art. 44, This relation M. Kummer has established by introducing certain conjugate complex numbers* 'I'(wi), "*'(=fr,)) ^(«^r,). ••• involving the * These complex numbers are defined as follows (see the memoir cited at the commencement of this article, sect. 3, and that in Crelle, vol. liii. p. 142) : — Let sr^ be a period satisfying the irreducible equation <i>{1sr^) = 0, and let a^, a be the incongruous roots of (f) (y) = 0, mod y ; 6,, 6j, . . . the remaining terms of a complete system of residues, mod q, so that (j) (6J, <f> (6,), ... are prime to q. Since ■er^" = or*,, mod q, and Vkq = Wj, we have, by Lagrange's indeterminate congruence (see Art. 10 of this Report), (=fi-«i) K-^s) • • • («^»-*i) {'<^k-K) . • . = 0, mod 2, or, since Wj — 6, divides <^ (5i) etc., <^(^) <^ (*J • • • K-«i) («f*-«2) • . . = 0, mod gr; i.e. (wj— a,) (wj— Oj) ... =0, mod q. We may now consider the n series of factors corresponding to the n values of k [the numbers a^, a,, ... are of course the same for two periods which satisfy the same irreducible equation, but not in general the same for any two periods], and, retaining among these factors only those which are different, we may take for *'(sr,) the complex number formed by combining as many of them as possible, in such a manner as to give a product which ia not divisible by q, but which is rendered divisible by q by the accession of any one factor not already contained in it. It is evident that *■ (•nr,) cannot contain all the factors ■oTj — aj, Wj — Oj ; let us then denote by itj— Mj a factor which is not contained in * (Wj) ; we thus obtain the relation * (isr,) (wj— tt^.) = 0, mod q, or, changing the primitive root oj into a/, ■♦•(or,.) {■aTrk — U/^) = 0, mod q. The conjugates of * (w,) are all complex numbers formed according to the same law as * (or,) itself; and, besides *(ori) and its conjugates, no other complex number can be formed according to that law. Also the number Mj which corresponds to a given period w^j is absolutely determined as soon as we have selected the multiplier *(tTj); for if two of the factors w^—a^, ■stj— Oj, ... were absent from * (w,) we should have *(x!7.) (wj-o,) = 0, *(CT,)(or4-Oj) = 0, mod q; and thence («i— «a) * (o^i) = 0, mod q, contrary to the hypothesis that a, and a, are incongruous, and that * («-,) is not divisible by q. The correspondence of the numbers Mj, m„ . . . , «,, with the periods «r,, CTj, . . . , «r„, can thus be fixed in as many ways as there are numbers conjugate to * (o7j), {. e. in • difi'erent ways. Art. 60.] EEPORT ON THE THEORY OF NUMBERS. 129 periods only, not themselves divisible by g, but each satisfying the n con- gruences included in the formula '^ {^r) i^kr - t^h) = 0, mod q, k = l, 2, 3, ..., n. From these congruences it is easy to infer that, if /(ta-,., war. •■•> ■o'»r) = be any identical relation subsisting for the periods, a similar relation f(ui, U2, ..., w„) = 0, mod q, wUl subsist for the numbers u^, u^, ... , «„ ; for we find '^{'^r)f{^T, •=^2r, •••) = ^K)/(«i> «2, ...). modg-, i.e.f{ui, U2, ...) = 0, modg-. Another important property of the complex num- ber "^ (wi) is that it is congruous to zero, mod q, for every one of the sub- stitutions isi = Ui, i!ri = M,.j, ■aii = u^^, ... except the first: thus the congruences ^ (tt,j) = 0, -^ (m,.j) = are satisfied, . . . but not ^ (uj) = 0, mod q. If, then, f(<e) be any complex number satisfying the congruence Sk {isT^Yf^w) = 0, mod 2™, but not the congruence ^ (:ir,)'» + i/(a,) = 0, mod g^ + S y(a)) is said to contain m times precisely the ideal factor of q corresponding to the substitution wj.r = Wj.. Since it can be shown that the numbers conjugate to ■^(isTi) are all different from one another, it follows from the definition, that the quotient ^{n) represents the number of conjugate ideal prime factors contained in the real prime q, appertaining to the exponent t. If 5' be a divisor of n, the definition of its ideal factors requires a certain modification, which we cannot here particularise. (See sect. 6 of M. Kummer's Memoir.) The two definitions, corresponding to the cases of q prime to n, and q a divisor of n, enable us, when they are taken together, to transfer to the general case when n is composite, the elementary theorems already shown to exist when n is prime (see Art. 47). We may add that it is easy to prove, in the general as in the special case (see Art. 48), that the number of classes of ideal nimibers is finite. 60. Application to the Theory of the Division of the Circle. — We cannot quit the subject of complex numbers without mentioning certain important investigations in which they have been successfully employed. The first relates to the problem of the division of the circle. In this problem the s 180 REPORT ON THE THEORY OF NUMBERS, [Art. 60. resolvent fiinction of Lagrange 2 d'xT (see Art. 30) is, as is well known, of primary importance. Retaining, with a slight modification, the notation of Art. 30, and still representing by X a prime divisor of ^-1, and by a a root of the equation = 0, let us consider the function F{a, x), which is a particular case of the resolvent, and let us represent the quotient ^:(j^^by+.W. We thus find [F(„,x)J= +,(a).+,(a)...<).,.,(.).J?«a:) (1) and in particular, observing that F{a, x) F{a^~^, x) =p, [F{a,x)Y=p^,{a).^,{a)...^^.,{a), (2) a result which is in accordance with the known theorem that [-F(a, x)]* is independent of x and is an integral function of a only. The resolution of the auxiliary equation of order X, the roots of which are the X periods of ^—r — roots of the equation — = 0, depends solely on the determination A X — -1. of the complex numbers 4'i(«)> ^2(«)> ••■> ^K-zip^- For when these complex numbers are known, we may equate F{a, x) to any X-th roots of the expression ^•vfr, (a).\|^2(«) ••• ■^x-2(«); from the value of F{a,x), thus obtained, those of F{a^,x), F{a^,x), ... may be inferred by means of equation (1); and, lastly, from the values of F{l,x), F{a,x), ..., F{a'^~^,x), the values of the periods themselves are deducible by the solution of a system of linear equations. To determine the numbers ^i(a), ^2i°)> ••• M. Kummer assigns the ideal prime factors of which they are composed, employing for this purpose the results cited in Art. 30. The equation >|^j (a) \//^j.(a~^)=p shows that >|'jfc(a) contains precisely ^{p — l) ideal prime divisors of ^, and no other complex prime. To distinguish the prime factors of p contained in •v/^j. (a) from those contained in \|/j(a-') M. Kummer avails himself of the congruence V. of Art. 30, viz., p—i Let X' = ^-- — , and u = y^', mod ^, so that u, M^ ...,u^~^ are the roots of A = 0, mod p ; also, to adapt the formulae of Art. 30 to our present purpose, let 6'^' = a, m = \', n = h\' ; it will result from these substitutions, that \|'jfc(u-*) = 0, mod^, if k and h satisfy the inequality [A] + [i-A]>X, where Art. 61.] REPORT ON THE THEORY OF NUMBERS. 131 [hi and [kK] are positive numbers less than X, and congruous, mod X, to h and kh respectively. If we represent by f{a) the ideal prime factor of p which appertains to the substitution a = u, this may be expressed by saying that ■v/'i (a) contains the factor /(a-*), if y^ + y^ > X, the symbols j- and j- denoting the least positive numbers satisfying the congruences ^a;=l, modX, and hx = k, mod X. Assigning, therefore, to the number h every positive value less than X compatible with this condition, we may write >!.,(«)= +a'n/(a-*), + a' being a simple unit which may be determined by the congruence ^k{a)= -1, mod(l-a)2*: it is not necessary to add a real complex unit, for a reason which has already appeared (see Art. 56, supra). From the expression for ■^/'^(a) a still simpler formula for F{a, xf" may be obtained, viz. m=\-\ ri-| [F{a,x)'Y=±a' n [/(a-'»)]UJ.t m = \ 61. Application to the Last Theorem of Fermiit. — The second investigation to which we shall advert relates to the celebrated proposition known as the 'Last Theorem of Format,' viz. that the equation a5" + ?/" = z" is irresoluble, in integral numbers, for all values of n greater than 2 J. As Fermat himself * The numbers 1/^^(0) are primary according to M. Kummer's definition (Art. 52) ; for F{a,x)F{o!',x) ^*('') = — PV^^)-= ' the summation extending to every pair of values of y^ and y^ that satisfy the congruence 7*1 + y "a = 1, mod/), in which y represents the same primitive root of/; that occurs in the expression F{a, x). Hence \//i(l)=i)-2= -1, modX, and >/'i(a) v//4(a-') = p = 1 = [x/^4(l)]^ mod A. Also V'i(o)— ^i(l) is divisible by (1 — a)'; for ^'*(l) = S{y^+%,) = i(l+*)(i'-l)(^-2)> observing that 1/, and y^ each receive all the values 1, 2, . . . , p—2 in succession. We have, therefore, the congruence ^\{^) = 0, mod X, from which it follows (see a note on the next article) that >/'j(a) = \//j(l), mod (1 — a)", or \/^j(a) = — 1, mod(l— a)'', as in the text, t Liouville, vol. xvi. p. 448. If. Kummer has also extended his solution of this problem to the case in which n is any divisor of /)— 1. See the memoir quoted in the last article, sect. 11. X Fermat's enunciation of this celebrated theorem is contained in the first of the MS. notes placed by him on the margin of his copy of Bachet's edition of Diophantus. It would seem that this copy is now lost ; but in the year 1670 an edition of Bachet's Diophantus was published at Toulouse, S 2 132 REPORT ON THE THEORY OF NUMBERS, [Art. 61. has left us a proof of the impossibility of this equation in the case of n = 4, by a method wliich Euler has extended to the case of n = 3, we may suppose, without by Samuel de Fermat (the son of the great geometei'), in which these notes are preserved (Diophanti Alexandrini Arithmeticorum libri sex, et de Numeris Multangulis liber unus, cum commentariis r. 0. Bacheti V. C. et observationibus D. P. de Fermat senatoris Tolosani. Tolosie 1670). The theorems contained in them are, with a few exceptions, enunciated without proof; and it may be inferred from the preface of S. Fermat that he found no demonstration of them among his father's papers. Nevertheless, in the case of several of these propositions, we have the assertion of Fermat himself that he was in possession of their demonstration; and although, when we consider the imperfect state of analysis in his time, it is surprising that he should have succeeded in creating methods which subsequent mathematicians have failed to rediscover, yet there is no ground for the suspicion that he was guilty of an untruth, or that he mistook an apparent for a real proof. In fact these suspicions are refuted, not only by the reputation for honour and veracity which he enjoyed among his contemporaries, and by the evidence of singular clearness of insight which his extant writings supply, but also by the facts of the case itself. {Gauss, vol. ii. p. 160, expresses himself unfavourably to Fermat: see especially p. 152.} It would be inexplicable, if his conclusions reposed on induction only, that he should never have adopted an erroneous generalization ; and yet, with the exception of the ' Last Theorem ' (the demonstration of which, after two centuries, is still incomplete), every proposition of Fennat's has been verified by the labours of his successors. There is, indeed, one other exception to this statement ; but it is an exception which proves the rule. In the letter to Sir Kenelm Digby which concludes the ' Commercium Epistolicum etc.' edited by Wallis (Oxford, 1658), Fermat euunciates the proposition that the numbers contained in the formula 2^+1 are all primes, acknowledging, however, that, though convinced of its truth, he had not succeeded in obtaining its demonsti'ation. This letter, which is undated, was written in 1658 ; but it appears, from a letter of Fennat's to M. de * * *, dated October 18, 1640, that even at that earlier date he was acquainted with the proposition, and had convinced himself of its truth (D. Petri de Fermat Varia Opera Mathematica, Tolosse, 1679, p. 162). It was, however, subsequently observed by Euler that 2'"+ 1 = 4294967297 = 641 x 6700417, i.e. that the undemonstrated proposition is untrue (Op. Arith. collecta, vol. i. p. 356). The error, if it is an error, is a fortunate one for Fermat ; it exemplifies his candour and veracity, and it shows that he did not mistake inductive probability for rigorous demonstration : — ' Mais je vous advoue tout net,' are his words in the letter last referred to, ' (car par advance je vous advertis que comme je ne suis pas capable de m'attribuer plus que je ne Sfay, je dis avec mSme franchise ce que je ne sfay pas), que je n'ay peu encore ddmonstrer I'exclusion de tons divi- seurs en cette belle proposition que je vous avois envoy^e, et que vous m'avez conferm6e touchant les nombres 3, 5, 17, 257, 6553, &c. Car bien que je reduise I'exclusion k la pluspart des nombres, et que j'aye mSme des raisous probables pour le reste, je n'ay peu encore d6monstrer n^cessairement la v6rit6 de cette proposition, de laqnelle pourtant je ne doute non plus k cette heure que je faisois auparavant. Si vous en avez la preuve assuree, vous m'obligerez de me la communiquer : car apr&s cela rien ne m'arrestera en ces matiferes.' The ' Last Theorem ' is enunciated by Fermat as follows : — ' Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaUter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere ; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.' (Fermat's DiophemtuB, p. 51.) Fermat has also asserted that neither the sum {ihid. p. 258) nor the difference {ibid. p. 338) of Art. 61.] REPORT ON THE THEORY OP NUMBERS. 133 loss of generality, that n is an uneven prime number X greater than 3, and we may write the equation in the symmetrical form x^ + y^ + z'^^O. The impos- two biquadrates can be a square. Each of these propositions comprehends the theorem that the sum of two biquadrates cannot be a biquadrate ; and of the second, we possess a very remarkable demonstra- tion by Fermat himself {ibid. p. 338; and compare Euler, E16mens d'Algfebre, vol. ii. sect. 13; Legendre, Th6orie des Nombres, vol. ii. p. I). The essential part of this demonstration consists in showing that, from any supposed solution of the Diophantine equation a;* — j/* = a square, another solution may be deduced in which the values of the indeterminates are not equal to zero, and yet are absolutely less than in the proposed solution, from which it immediately follows that the Diophantine equation is impossible. This method has been successively employed by Euler {loc. cit.) to demonstrate several negative Diophantine propositions, and in particular the theorem that the sum of two cubes cannot be a cube. The only arithmetical principles (not included in the first elements of the science) which are employed by Euler and Fermat in their applications of this method, relate to certain simple properties of the quadratic forms x' + y", a^ + 2y', ai'+Zy^ ; and as these principles seem inadequate to overcome the difficulties presented by the equation x" + y" + a" = 0, when n is > 4, it is probable that Fermat's ' demonstratio mirabilis sane ' of the general theorem was entirely different from that which he has incidentally given of the particular case. The impossibility of the equation x" + y" + s" = for n = 5 was first demonstrated by Legendre (M^moires de I'Acad^mie des Sciences, 1823, vol. vi. p. 1, or Th^orie des Nombres, vol. ii. p. 361. See also an earlier paper by Lejeune Dirichlet, Crelle, vol. iii. p. 354, with the addition at p. 368, and a later one by M. Lebesgue, Liouville, vol. viii. p. 49); for n = 14, by Dirichlet (Crelle, vol. ix. p. 390); and for « := 7, by M. Lam6 (M6moires des Savans Etrangers, vol. viii. p. 421, or Liouville, vol. v. p. 195. See also the Comptes Rendus, vol. ix. p. 359, and a paper by M. Lebesgue, Liouville, vol. V. pp. 276 and 348). But the methods employed in these researches are specially adapted to the particular exponents considered, and do not seem likely to supply a general demonstration. The proof in Barlow's Theory of Numbers, pp. 160-169, is erroneous, as it reposes (see p. 168) on an elementary proposition (cor. 2, p. 20) which is untrue. A memoir by M. Kummer on the equation SB'^+y"* = s'*, in which complex numbers are not employed, and in which no single case of the theorem is demonstrated (Crelle, vol. xvii. p. 203), is nevertheless of great interest for the number of auxiliary propositions contained In it. Of the same character are the notes by MM. Lebesgue and Liouville, in Liouville's Journal, vol. v. pp. 184 and 360, and a few theorems given without demon- stration by Abel, CEuvres, vol. ii. p. 264. Li the year 1847, M. Lam6 presented to the Academy at Paris a memoir containing a general demonstration of Fermat's Theorem, based on the properties of complex numbers (Comptes Rendus, vol. xxiv. p. 310; Liouville, vol. xii. pp. 137 and 172). It was, however, observed by M. Liouville (Comptes Rendus, vol. xxiv. p. 315), that this demonstration is defective, as it assumes, without proof, the proposition that a complex number can be represented, and in one way only, as the product of powers of complex primes — a proposition which, as we have seen, is untrue, unless we admit ideal as well as actual complex primes. The discussion on M. Lamp's memoir attracted Cauchy's attention to Fermat's Theorem; and the 24th and 25th volumes of the Comptes Rendus contain several communi- cations from him on the subject of complex numbers [or polyiiomes radicaux, as he has preferred to term them]. In the earlier papers of this series, Cauchy attempts to prove a proposition which, as we have already observed (see Art. 41), is untrue for complex numbers considered generally, viz. that the norm of the remainder in the division of one complex number by another can be rendered less than the norm of the divisor (see Comptes Rendus, vol. xxiv. pp. 517, 633, and 661). Elsewhere {ibid. p. 579) he 134 REPORT ON THE THEORY OF NUMBERS. [Art. 61. sibility of solving this equation has been demonstrated by M. Kummer, first, for all values of X not included among the exceptional primes * ; and secondly, for all exceptional primes which satisfy the three following conditions : — (1) That the first factor of H, though divisible by X, is not divisible by X« (see Art. 50). (2) That a complex modulus can be assigned, for which a certain definite complex unit is not congruous to a perfect X-th power. (3) That Bif^ is not divisible by X', B^ representing that Bemoullian number [ic ^ m — 1] which is divisible by X f. Three numbers below 100, viz. 37, 59, 67, are, as we have seen, exceptional primes. But it has been ascertained by M. Kummer that the three conditions just given are satisfied in the case of each of those numbers ; so that the impossibility of Fermat's equation has been demonstrated for all values of the exponent up to 100. Indeed, it would probably be difficult to find an exceptional prime not satisfying the three conditions, and consequently excluded from M. Kununer's demonstration. We must confine ourselves here to an indication of the principles on which the demonstration rests in the case of the non-exceptional primes J. assumes the proposition as a hypothesis, and deduces from it conclusions which are erroneous (pp. 581, 582). But at p. 1029 he recognises and demonstrates its inaccuracy. The results at which he arrives in his subsequent papers on the same subject are, for the most part, comprehended in M. Rummer's general theory (Comptes Eendus, vol. xxv. pp. 37, 46, 93, 132, 177). In one place, however (p. 181), he enimciates, though without demonstrating, the following important result : — 'If the equation ac'^ + j/^+a'' = be resoluble, x, y, z denoting integral numbers prime to X, the Bum l^-4 + 2^-* + 3^-»+...+{KX-l)}^-' is divisible by X.' (Compare M. Rummer's memoir in the Berlin Transactions for 1857, p. 64.) The investigation of the Last Theorem of Fermat has been twice proposed as a prize-question by the Academy of Paris — first at some time previous to 1823 (see Legeudre's memoir already cited, in vol. vi. of the M^moires de FAcad^mie des Sciences, p. 2), and again in 1850 (Comptes Eendus, vol. XXX. p. 263) : at neither time was the prize adjudged to any of the memoirs received. On the last occasion, after several postponements of the date originally fixed for the award, the prize was ultimately, in 1857 {ib. vol. xliv. p. 158), conferred on M. Rummer, who had not been a competitor, for his researches on complex numbers. * Liouville, vol. xvi. p. 488, or Crelle, vol. xl. p. 131. t See the memoir No. 15 in the list of Art. 41. + When X is not an exceptional prime, the equation a^ + y^-l-a^ = is irresoluble not only in ordinary integral numbers, but also in any complex integers composed of X-th roots of unity. The demonstration does not possess the same generality when \ is an exceptional prime satisfying the three conditions cited in the text. In this case M. Rummer has only shown that the equation Art. 61.] REPORT ON THE THEORY OF NUMBERS. 135 We may suppose that X is greater than 3, and that no two of the numbers X, y, z admit any common divisor. And first, let none of them be divisible by \-a, a still representing a root of the equation = 0. Since for x we may write a'x, we may assume that x, y, z are of the form x = a + {l-ayX, y = h + {l-ayY, z = c+{i-ayz, a, b, c denoting integral numbers prime to X, which evidently satisfy the con- gruence a + b + c = 0, mod X. The equation a;^ + ?/^ + z^ = may then be written ^""S (x + ay) (x + a^y) {x + a^y) (x + a^~^y) = — z\ No two of the factors of which the left-hand member is composed can have any common divisor; each of them is therefore the product of a perfect X-th power by a unit; so that we may write, x + a'y^aPeiaJT^, e{a) denoting a real unit. Since v^ is an actual number, it follows (remembering that X is not an exceptional prime) that v is also actual ; hence ^ is congruous, mod X, to a certain integral number m. Eliminating my.e{a) between the two congruences x + a'y^maPeia), and a; -I- a"*?/ = ma"Pe (a), mod X, ^e find a-i'{x + a'y)-al'{x + a-'ij) = 0, mod X. For the modulus (1 — a) this congruence is identically satisfied *. That it should be satisfied, mod (1 — a)^, we must have the relation (a + &) p = hs, mod X ; whence, putting ^ = h, mod X, we have p = ks, mod X. Substituting this value for p, we find that the congruence a~** (ic + a'y) — a'" {x + a~'y) = is identically satisfied, mod (1 — a)^ ; but in order that it should be satisfied, mod (1 — a)*, we have the condition sn{2k-l)(k-l)-3s{{k-l)y" + kx"} = 0,mod\ a;* + y^ + s^ = is irresoluble when we suppose that x, y, z are ordinary integral numbers prime to A , or else complex numbers containing the binary periods a + o~', one of which has a common divisor with A. * Since A is divisible by (1— a)''"', and since <^ («) = </> (1) + (a - 1) </''(l) + (a - 1)' y^ + • • • , it is readily seen that, if r ^ A— 1, the conditions for the divisibility of <^ (a) by (1 —a)' are «/) (1) = 0, <^'(1) = 0, . . . , </)('•-') (1) = 0, mod A. 136 REPORT ON THE THEORY OF NUMBERS. [Art. 61. where x" and y" are the values (for a = 1) of the second derived functions of X and y with respect to a. This conditional congruence must be satisfied for every value of s; either therefore A; = l, mod X, or 2^=1, mod X. The supposition i = 1 is inadmissible ; for it implies that a = 0, mod X, contrary to the hypothesis. Hence we must have 2^=1, and a = 6, or, by parity of reasoning, a^h^c, mod X. But also a + & + c = 0, mod X, whence we again infer the inadmissible conclusion a = fc = c = 0, mod X. Secondly, let one of the numbers x, y, z (for example, z) be divisible by 1 — a ; it will be convenient to consider the equation in the generalised form «^ + 2^ = ^(a)(l-a)'»^z\ (1) in which x, y, and z are all prime to 1 — a, and E (a) is any unit. We may assume that the values of x and y are of the form x = a-\-{l — a)-X, y=b + {l-ayY, a and 6 being prime to X, but satisfying the relation a + h = 0, mod X. In the first place, m must be greater than 1. For since a^ = a\ and y^ = h'', mod (1 — a)^ + \ i£ x^ + y^he divisible by (1 — a)\ a^ + h^ is divisible by X^, and therefore s^ + y*" by (1 — a)^+'. Again, each of the factors x + ay, x-\-a^y, ..., x + a^~^y is divisible once, and once only, by 1 — a ; whence it follows that x + y \s divisible by (1 — 0)*"^"*^ + ^ and that no two of the X factors of x^ + y*- have any other common divisor than 1 — a. Hence the X factors x + y x + ay x + a^~^y (l-a)'»^-^ + i' IT^' ' 1-a are relatively prime, and may be represented by expressions of the form eo(«)0o\ ei(a)^i\ , e;,_i(a)<^J_i, Co (a), ei(a), ... representing units, and ^^, <pi\ ... X-th powers prime to 1 — a. Eliminating x and y from the three equations «+ 3/ = e„(a)</>o^(l-o)"'^-^ + ', a; + o''y = e,(a)</)/(l-a), X + a'y = e,(a)<pj^(l-a), we obtain a result of the form <^/ + e(«)<^.^ = ^i(«)(l-a)('»-i)^.^o\ • (2) e(a) and Ei(a) denoting two units. But, as in the former case, it may be shown that ^/ and ^/ are congruous, mod X, to real integers, and Art. 62.] EEPORT ON THE THEOEY OF NUMBERS. 137 (1 — a)<"'-i>^ = 0, mod X, because m>l. Hence e(a) is also congruous to a real integer for the modulus X, and is therefore a perfect X-th power by a property of every non-exceptional prime (see Art. 52). The equation (2) therefore assumes the form ^^x ^ y\ ^ ^^ („) ^j^ ^^ _ «)(™-i)x K, therefore, the proposed equation (1) be possible, it will follow, by successive applications of this reduction, that the equation is also possible. But this equation has been shown to be impossible ; the equation (1) is therefore also impossible. 62. Application to the Theory of Numerical Equations. — In the Monats- berichte for Jime 20, 1853 (see also the Monatsberichte for 1856, p. 203), M. Kronecker has enunciated the following theorem : — ' The roots of any Abelian equation, the coefficients of which are integral numbers, are rational functions of roots of unity.' The demonstration of this theorem (Monatsberichte for 1853, pp. 371-373) depends on a comparison of a certain form, of which the resolvent function of any Abelian equation is susceptible, with M. Kummer's expression for the resolvent function in the case of the equation of the division of the circle (see Art. 60). It thus involves considerations relating to ideal numbers. Two propositions of a more special character, and closely connected with one another, have also been given by M. Kronecker (Crelle, vol. liii. p. 173). Their demonstration is immediately deducible from the principles of Dirichlet's theory of complex units : — ' If unity be the analytical modulus of every root of an equation, of which the first coefficient is unity and all the coefficients are integral numbers, the roots of the equation are roots of unity.' ' If all the roots of an equation (having its first coefficient unity and all its coefficients integral) be real and inferior in absolute magnitude to 2, so that they can be represented by expressions of the form 2 cos a, 2 cos j8, 2 cos 7, . . . the arcs a, (8, y are commensurable with the complete circumference.' In the following proposition M. Kronecker has extended a theorem of M. Kummer's (Art. 42) relating to complex units composed with roots of unity of which the index is a prime, to complex units composed with any roots of imity (Crelle, vol. liii. p. 176) : — ' Every complex unit composed with the roots of the equation «>" = 1, can be rendered real by multiplication with a 4«-th root of unity. If n be even, 138 KEPORT ON THE THEORY OF NUMBERS. [Art. 63. a 2n-th root will always suffice ; and if n be a power of a prime, an n-th root will suffice.' The demonstration of this proposition is also deducible from Dirichlet's principles. 63. Tables of Complex Primes. — In M. Kumraer's earliest memoir on com- plex numbers (Liouville, vol. xii. p. 206) he has given a table of the complex factors, composed of X-th roots of unity, which are contained in real primes of the form mX + 1 inferior to 1000, X representing one of the primes 5, 7, 11, 13, 17, 19, 23. This memoir was written before M. Kummer had considered the complex factors of primes of linear forms other than m\ + l, and before he had introduced the conception of ideal numbers. The complex prime factors of real primes of those other linear forms are, therefore, not exhibited in the Table; and the live numbers of the form 23m + 1, 47, 139, 277, 461, 967, each of which contains 22 ideal factors composed of 23rd roots of unity, are repre- sented as products of 11 actual factors (each of which contains two reciprocal ideal factors). The tentative methods by which the complex factors were dis- covered are explained in sect. 9 of the memoir cited. Since the full develop- ment of M. Kummer 's theory, Dr. Reuschle has undertaken to complete and extend the Table. He has already given tables containing the complex prime factors of all real primes less than 1000, composed of 5th, 7th, 11th, 13th, 17th, 23rd, and 29th roots of unity, together with the complete solution of the congruences corresponding to the equations of the periods (see the Mo- natsberichte for 1859, pp. 488 and 694, and for 1860, pp. 150 and 714). For 5, 7, 11, 13, 17, the complex primes are exhibited in a primary form; for 19, 23, and 29 they are exhibited in a form which satisfies the condition /(a)=/(l),mod(l-«)', but not the condition /(a)/(a-i) = [/(I)]', mod X. The ideal factors Dr. Reuschle represents by their lowest actual powers ; for 23 this power is the cube, for 29 it is the square ; for 11, 13, 17, 19, as well as for 5 and 7, all complex prime factors of real primes less than 1000 are actual. It appears from the Table (and it has indeed been proved by M. Kum- mer), that 29 is an ' irregular determinant ' (see Art. 49, note) ; for the number of classes is 8, while the square of every ideal number (occurring as a factor of a real prime inferior to 1000) is actual. The methods employed by Dr. Reuschle in the calculation of his tables have not yet been published by him. In some instances, as M. Kummer has observed, they have not led him to the simplest possible forms of the ideal primes. Art. 64.] REPORT ON THE THEORY OF NUMBERS. 139 A particular investigation relating to the ideal factors of 47, composed of 23rd roots of unity, has been given by Mr. Cayley (Crelle, vol. Iv. p. 192, and IvL p. 186). 64. The investigations relating to Laws of Reciprocity, which have so long occupied us in this report, have introduced us to considerations apparently 80 remote from the theory of the residues of powers of integral numbers, that it requires a certain effort to bear in mind their connexion with that theory. It wUl be remembered that the complex numbers to which our attention has been directed are not of that general kind to which we have referred in Art. 41, but are exclusively those which are composed of roots of unity. The theory of complex numbers, in the widest sense of that term, does indeed present to us an important generalisation of the theory of the residues of powers ; for the theorem of Format (see Art. 53) subsists ahke for every species of com- plex numbers. But the complex numbers of Gauss, of Jacobi, and of M. Rum- mer force themselves upon our consideration, not because their properties are generalisations of the properties of ordinary integers, but because certain of the properties of integral numbers can only be explained by a reference to them. The law of quadratic reciprocity does not, as we have seen, necessarily require for its demonstration any considerations other than those relating to ordinary integers ; the real prime numbers of arithmetic are here the ultimate elements that enter into the problem. But when we come to binomial congruences of higher orders, we find that the true elements of the question are no longer real primes, but certain complex factors, composed of roots of unity, which are, or may be conceived to be, contained in real prunes. For we find that the law which expresses the mutual relation (with respect to the particular kind of congruences considered) of two of these complex factors is a primary and simple one ; while the corresponding relations between the real primes them- selves are composite and derivative, and, in consequence, complicated. It thus becomes indispensable, for the investigation of the properties of real numbers, to construct an arithmetic of complex integers ; and this is what has been accomplished by the researches, of which an account has been given in the preceding articles. The higher laws of reciprocity (like that of quadratic residues) may be considered as furnishing a criterion for the resolubUity or irresolubUity of binomial congruences ; and this, though not the only application of which they are susceptible, is that which most naturally suggests itself. When the binomial congruence is cubic or biquadratic, it is easy to resolve the real prime modulus T 2 140 REPORT ON THE THEORY OF KUMBERS. [Art. 65. into factors of the form a + hp, or a + hi (Arts. 37 and 24), and equally easy to determine the value of the critical symbol of reciprocity by a uniform and elementary process (see Art. 36). For these, therefore, as well as for quadratic congruences, the criterion deducible from the laws of reciprocity is all that can be desired. But for binomial congruences of higher orders this criterion is not a satisfactory one, because of the difficulty of obtaining the re.solution of a real prime into its complex factors, and also because of the impossibility of determining the value of the critical symbol by the conversion of an ordinary fraction into a continued fraction. The only known criterion applicable to such congruences is the following, the demonstration of which is deducible from the elements of the theory of the residues of powers : — Let a;" = ^, mod p, represent the proposed congruence ; it will be resoluble or irresoluble according as the index of A is or is not divisible by d, the gi'eatest common divisor of ft andp — 1, i.e. according as the exponent «-l to which A appertains is or is not a divisor of ^ , (see Arts. 14 and 15). 65. Solution of Binomial Congruences. — We now come to the problem of the actual solution of binomial congruences — a subject upon which our know- ledge is confined within very narrow limits. When a table of indices for the prime p has been constructed, the reso- lution of every binomial congruence, if it be resoluble, or, if not, the demon- stration of its irresolubility, is implicitly contained in it. But to use a table of indices for the solution of a binomial congruence is, as we have already observed in a similar case (Art. 16), to solve a problem by means of a recorded solution of it. When the congruence x^ = A, mod j?, is resoluble, its solution may always be made to depend on that of a congruence of the form oc^ = a, mod p, where d is the greatest common divisor of n and p — t, and where a = A', mod p, and ns = d, mod^ — 1. We may therefore suppose that, in the congruence x" = A, mod p, n is a divisor of p — 1. This congruence (if re- soluble at all) will have as many roots as it has dimensions ; if ^ be any one of them, and 1, 6^, 0^, ..., 6„_i be the roots of the congruence x" = l, mod p, the roots of a;"=^, mod p, will he ^, ^d^, ^62, ..., ^d„_i; so that the complete resolution of the congruence x'*=A, mod p, requires, first, the determination of a single root of that congruence itself, and, secondly, the complete resolution of the congruence a;" = 1, mod p. With regard to the first of these requisites, in the important case in which the exponent t to which A appertains is prime to n, a value of x satisfying the congruence a;"=^, mod^, can be determined Art. 66.] EEPORT ON THE THEOBY OP NUMBERS. 141 by a direct method (Disq. Arith., Arts. 66, 67). For, in this case, it will always happen that one value of « is a certain power A'' of A, where k is determined by the congruence kn = l, mod t. Nor is it necessary, in order to determine k, to know the exponent t to which A appertains ; it is sufficient to have ascertained that it is prime to n ; for, if we resolve p — 1 into two factors prime to one another, and such that one of them is divisible by n and contains no prime not contained in n, the other will be divisible by t, and may be employed as modulus instead of t in the congruence kn = l, mod t. When this method is inapplicable, we can only investigate a root of the congruence x'' = A, mod^ (where A is different from 1), by tentative processes, which, however, admit of certain abbreviations (Disq. Arith., Arts. 67, 68). The work of Poinsot (Reflexions sur la Theorie des Nombres, cap. iv. p. 60) contains a very full and elegant exposition of the theory of binomial congru- ences ; but neither he nor any other writer subsequent to Gauss has been able to add any other direct method to that which we have just mentioned. 66. Solution of the Congruence a;" = 1, mod p. — When a single root of the congruence a;" = ^ is known, we may, as we have seen, complete its resolution by obtaining all the roots of the congruence a;'* = l, mod j9. The methods of Gauss, Lagrange, and Abel for the solution of the binomial equation ic" — 1 = are in a certain sense applicable to binomial congruences of this special form. It is evident, from a comparison of several passages in the Disquisitiones Arithmeticae *, that Gauss himself contemplated this arithmetical application of his theory of the division of the circle, and that he intended to include it in the 8th section of his work, which, however, has never been given to the world. In fact, the method of Abel t which comprehends that of Gauss, and which gives the solution of any Abelian equation, is equally applicable to any Abelian congruence ;' i.e. to any completely resoluble congruence of order m, the m roots of which (considered with regard to the prime modulus p) may be represented by the series of terms r, <j>{r), (}>'{r), ..., cf>">-^(r), the symbol <p denoting a given rational [fractional or integral] ftmction. And as we can always express the roots of an Abelian equation by radicals {i.e. by * See Disq. Arith., Arts. 61, 73, and especially Art. 335. t See Abel's memoir, ' Sur une classe particulifere d'6quations r^solubles alg^briquement,' sect. 3. ((Euvres, vol. i. p. 114, or C'relle, vol. iv. p. 131), and M. Serret's Algfebre Sup6rieure, 26th and 27th lessons. 142 REPORT ON THE THEORY OF NXJMBERS. [Art. 66 the roots of equations of two terms), so also the solution of an Abelian con- gruence depends ultimately on the solution of binomial congruences. When, for any prime modulus, an Abelian equation admits of being considered as an Abelian congruence, so precise is the correspondence of the equation and the congruence, that (as Poinsot has observed in a memoir in which he has occupied himself with the comparative analysis of the equation a;" = l, and the congruence x" = l, modp*) we may consider the analytical expression of the roots of the equation as also containing an expression of the roots of the congruence ; and by giving a congruential interpretation t to the radical signs which occur in that expression, we may elicit from it the actual values of the roots of the congruence. An example taken from Poinsot's memoir x' — 1 will render this intelligible J. The six roots of the equation — = are comprised in the formula where the signs + and — are to be successively attributed to ^ — 7, and where the product of the two cube roots is +^ — 7, or -^ - 7, according to the sign attributed to >/ — 7. Considering the equation as a congruence with regard to the modulus 43, and observing that V^^ = ± 6, mod 43, ^21 = + 8, mod 43, we obtain in the first place x= | + l4/T6 + |4/^mod43, and a;=_^ + 13/22 + 1^^ mod 43, the product of the two cube roots being congruous to + 6 in the first formula, and to — 6 in the second ; and finally, observing that * ' Sur I'Application de I'Algfebre k la Tli^orie des Nombres,' Memoires de I'Acadtoiie des Sciences, vol. iv. p. 99. / — '-'' *^ X ■ t Gauss employs the symbol y/A, mod p, to denote 4 root of the congruence x" = A, mod p, just as he employs the symbol — , mod p, to denote the root of the congruence Ax = B, mod p. The conffruential radical is/ A, mod p, has of course as many values as the congruence x" = .4, mod p, has solutions ; if that congruence be irresoluble, the symbol is impossible. X See the memoir cited above, p. 125. Art 66.] REPORT ON THE THEORY OF NUMBERS. 143 4/16 = 21, - 4/3^= 14, - 4/22 =-15, - 3, - 18, mod 43, 2, -12, mod 43, 4, 19, mod 43, + 11, mod 43, 4/-2=+ 9, -20, and attending to the limitation to which the cube roots are subject, x=-%, +11, +21, or, -2, +4, +16; mod 43. Thus the complete solution of a congruence of the sixth order is obtained by- means of binomial congruences of the second and third orders only. An essential limitation to the usefulness of this method arises from the circumstance that it does not always (or even in general) happen that (as in the example just given) each surd entering into the expression of the root becomes separately rational. For that expression may itself acquire a rational value, while certain surds contained in it continue irrational, precisely as, in the irreducible case of cubic equations, a real quantity is represented by an imaginary formula. To illustrate this point by an example, let us consider the same congruence the expression X' X- = with respect to the modulus 29 *. Here in U' - 1 v^^^v/^r^ hi'- U'^-i^\ x= f + where p denotes a cube root of unity, we have, putting ^ — 7 = + 14, and p = l, the irrational cube roots disappearing of themselves. Again, putting we find .--7, 1 - /I = 7 + (7)*=7 + 16=-6 or -9, where every radical becomes rational of itself Similarly taking the values J^=-U, p^h-l + ^/Z-3), we find x= —5 or — 13. But lastly, putting ^ — 7 = — 14, /> = 1, we find a; = 12 + J [14 + 7^2]* + J [14 -7^2]*. * Ibid. p. 132. 144 REPOET ON THE THEORY OF NUMBERS. [Art. 66. To rationalise this expression, we have to observe that 14 + 7^^, relatively to the modulus 29, is the cube of a complex number of similar form ; in fact, we have (14±7;^)=(5 + 11^2)\ mod 29, whence x= -4. To elicit, therefore, the value of this root from the iiTational formula, we are obliged to solve the cubic congruence a;' = 14 + 7^2, which, although of lower dimensions than the proposed congruence, is probably less easy to solve tentatively, because 29 has 29' - 1 = 840 residues of the form a + hj2, and only 29 - 1 = 28 ordinary integral residues ; so that practically the method fails. Theoretically, however, the relation between the analytical expression of the equation-roots and the values of the congruence-roots is of considerable importance, and the subject would certainly repay a closer examination than it has yet received. We may add that, if m be a divisor of j» — 1, the complete solution of an Abelian con- gruence of order m requires only two things, — 1st, the complete solution of the congruence aj™ — 1 = 0, mod j?, and, 2ndly, the determination of a single root of a certain congruence of the form x^ — a = 0, mod^, in which a is an ordinary integer; so that in this case (which is that of the congruence x' — 1 -r =0, mod 43) we obtain a real, and not only an apparent reduction of the proposed congruence *. It should also be observed that the primitive roots of the equation a;" — 1 — — - = furnish, when rationalised, the primitive roots of the congruence K" — 1 J = 0, mod p. This, the only direct method that has ever been suggested for the determination of a primitive root, appears to be the same as that referred to by Gauss in the Disq. Arith. (Art. 73). Poinsot expresses the conviction that this method of rationalisation is applicable to any congruence corresponding to an equation, the roots of which can be expressed by radicals f. With regard to equations of the second, third, and fourth orders this is certainly true. If, for example, the biquadratic equation Fi{x) = (i be completely resoluble when considered as a congruence * This will be at once evident, if we observe that when the congruence »" = 1, mod p, is completely resoluble, its roots may be employed to replace, in Abel's method, the roots of the equation a!""— 1 =0, t See the memoir cited above, p. 107, and M. Libri, Memoires de Math6matique et Phy- sique, p. 63. Art. 6?.] REPORT ON THE THEORY OF NUMBERS. 145 for the modulus^, so that F^ (ic) = {x- ai) (x - a.^ {x - a^) {x - a^), mod p, it is plain that the four roots of F(x) = 0, and the four numbers tti, a^, a^, a^^ may be obtained by substituting, in the general formula which expresses the root of any biquadratic equation as an irrational function of its coefficients, the values of the coefficients of the functions F (x) and (x — ai) (x — a^ {x — a^ {x — a^ respectively. But these two sets of coefficients differ only by multiples of p ; i.e. the values of aj, a.,, a^, a^ can be deduced from the expressions of the roots of F(x) = by adding multiples of p to the numbers which enter into those expressions. But this reasoning ceases to be applicable to equations of an order higher than the fourth, because no general formula exists repre- senting the roots of an equation of the fifth or any higher order. If, therefore, F{x) = be an equation of the nth order, the roots of which can be expressed by a radical formula, and which is also completely resoluble when considered as a congruence for the modulus p, so that F(x) = (x — Oi) {x — a^) ... {x — a„), mod p, it will not necessarily follow that the formula which gives the roots of i^(a;) = is also capable (when we add multiples of p to the numbers contained in it) of giving the roots of (x-Oi) (x-tta) ... (x-a„) = 0, i.e. the roots of the congruence F{x) = 0, modjp; and thus the principle enunciated by M. Poinsot is, it would seem, not rigorously demonstrated. 67. Cubic and Biquadratic Congruences. — The reduction of cubic con- gruences to binomial ones has been treated of by Cauchy (Exercices de Mathe- matiques, vol. iv. p. 279), and more completely by M. Oltramare (Crelle, vol. xlv. p. 314). Some cases of biquadratic congruences are also considered by Cauchy in the memoii- cited, p. 286. The following criteria for the resolubility or irresolubility of cubic congruences include the results obtained by M. Oltramare, I. c, and appear sufficiently simple to deserve insertion here : — Let the given cubic congruence be ae^ + 3be^ + 3cd + d = 0, modp, p denoting a prime greater than 3, which does not divide the discriminant of the congruence ; i.e., the number D= -a^d^ + 6ahcd-4:ac^-4^d¥ + 3b'c^; V 146 REPORT ON THE THEORY OF NUMBERS. [Art. 68. and, in connection with the congruence, consider the allied system of functions • U = (a, h, c, d) (x, yY, H= {ac - b\ is(ad - he), hd - c^) (x, yY, 4>=(-a2rf + 3a6c-263, -abd + 2ac^ + b^c, acd-2¥d + hc^, ad^-3bcd + 2c^)(x,yy, which are connected by the equation let also « and denote the values of U and $ corresponding to any given values of x and y, which do not render H=0, mod p. Then, if ( — ) = — 1, the congruence has always one and only one real root; if ( )= +1> it has either three real roots, or none : viz., if ^?Ai '^ = + 1, it has three ; if ^?ii ~ — 1^ =p, or =p^, it has none. The interpretation of the cubic symbol of reciprocity will present no difficulty if we observe that y/ — D, modp, is a real mteger if p = 3n + l, i.e. if (- — ) = lj and that, if ^ = 3w-l, i.e. if ( j= —1, we have J^D= J~^ X s/¥D = {p - p') x/p, mod p, so that \/ — D, mod^, is a complex integer involving p. It will however be observed that the application of the criterion requires in either case the solu- tion of a quadratic congruence, r^= —D, modp, or r^=\D, mod^. Similar, but of course less simple, criteria for the resolubility or irresolu- bility of biquadratic congruences may be deduced from the known formulae for the solution of biquadratic equations. 68. Quadratic Congruences — Indirect Methods of Solution. — The general form of a quadratic congruence is ax* + 2 6a; + c = 0, mod p, where p denotes an uneven prime modulus, and a is a number prime to p. It may be immediately reduced to the binomial form r* = D, mod p, by putting r=ax-\-b, D=b^ — ac, mod p. The number of its solutions is 2, 0, or 1, according as Z) is a quadratic residue * See a note by Mr. Cayley in Crelle's Jaurnal, vol. 1. p. 285. Art. 68.] REPORT ON THE THEORY OF NUMBERS. 147 or non-residue o£ p, or Is divisible by^, and is therefore in every case expressed by the formula 1 + (—) ■ If jp = 4n + 3, and (— ) = 1, the congruence r^-D=0, mod.p, is satisfied by r=D"^^, and rs— Z>" + ', and is in fact resoluble by the direct method of Art. 6"). But no direct method, applicable to the case when p = 4:n + l, is at present known. Two tentative methods are proposed in the sixth section of the Disquisitiones Arithmeticse. They are both applicable to congruences with composite as well as with prime modules. This circumstance is important, because, when the modulus is a very great number, we may not be able to tell whether it is prime or composite, and, if composite, what the primes are of which it is composed, although, when the prime divisors of a composite modulus are known, it is simplest first to solve the congruence for each of them separately, and afterwards (by a method to which we shall hereafter refer) to deduce from these solutions the solution for the given composite modulus. To apply the first of Gauss's methods, the congruence is written in the form r^ = D + Py, P denoting the modulus. If in the formula V=D-]- Py we substitute for y in succession all integral values which satisfy the inequality — -p <y <\P — ^ , and select those values of V which are perfect squares, their roots (taken positively and negatively) will give us all the solutions of the congruence. We should thus have I{\P) or 1 + /(^P) trials to make, / denoting the greatest integer contained in the fraction before which it is placed. If, however, we take any number E, greater than 2, and prime to P (it is simplest to take for E a prime, or power of a prime), of which the quadratic non-residues are a,h,c, ..., and then determine the values of a, ;8, 7, ... in the congruences a = D + aP, mod E, h = D-\-^P, mod E, &c., we shall find that every value of y contained in one of the linear forms mE + a, mE+l3, &c., gives rise to a value of V which is a quadratic non-residue of E, and which cannot, therefore, be a perfect square ; so that we may at once exclude these values of y from the series of numbers to be tried. A second excludent E' may then be taken, and by its aid another set of linear forms may be determined, such that no value of y contained in them can satisfy the con- gruence. Thus the number of trials may be diminished as far as we please. The application of this method is still further facilitated by the circumstance that it is not necessary actually to solve the congruences a = D-^aP, mod E, ... u 2 148 REPORT ON THE THEORY OF NUMBERS. [Art. 68- but only the single congruence D + Py = 0, mod JF (Disq. Arith., Art. 322). Gauss's second method depends on the theory of quadratic forms ; it supposes that the congruence is written in the form r* + Z)=0, mod P. By a tenta- tive process (abbreviated, as in the first method, by the use of excludents) Gauss obtains all possible primitive representations of P by the quadratic forms of determinant — D ; whence the complete solution of the congruence r^ + D = 0, mod F, is immediately deduced. This method involves the con- struction of a complete system of quadratic forms of determinant —D, or, if the prime factors of Z) be known, of one genus of forms of that system ; it becomes therefore more difficult of application as D increases, whereas the first method is not afiected by the increase of D. The second method, how- ever, especially recommends itself when P is a very great number; in fact, if we do not employ any excludent, the number of trials required by the first method varies (approximately, and when P is a great niunber) as P, whereas, on the same supposition, the number of trials required by the second method varies as v25 x \/P. M. Desmarest (in his Thdorie des Nombres) has proposed a method less scientific in its character than those of G«.uss, but sometimes easUy applicable in practice. He has shown that if the congruence r'^-\-D = Q, mod P, be resoluble, we can always satisfy the equation mP = x'^-\- Dy'^ with a value. of P m inferior to r^ + 3, and of y not superior to 3. The demonstration of this theorem is not very satisfactory, and the number of trials that it stUl leaves is very great, viz. 3 (l (— ) -1- 3j • The application of Gauss's second method is rendered somewhat more uniform, and at the same time the necessity for constructing a system of quadratic forms of determinant —D is avoided by the following modification of it : — By a known property of quadratic fonns, whenever the congruence r^ + D=0,inodP, is resoluble, the equation mP = x^ + Dy^ is resoluble for some value of m<2y/^D. By assigning, therefore, to m all values in suc- cession which are inferior to that limit, and which satisfy the condition (yJ = (x^), and then obtaining (by Gauss's method) all prime representations of the resulting products by the form x^ + Dy^, we shall have x' x" r= + -,, r= + — ,.,., mod P, Art. 69.] REPORT ON THE THEORY OP NUMBERS. 149 x, y', x", y", etc. denoting the different pairs of values of x and y in the equation mP = x'^-\-Dy^. 69. General Theory of Congruences. — We may infer fi-om several passages in the Disquisitiones Arithmeticse, that Gauss intended to give a general theory of congruences of every order in the 8th section of his work, and that, at the time of its puhlication, he was already in possession of the prin- cipal theorems relating to the subject *. These theorems were, however, first given by Evariste Galois t, in a note published in the Bulletin de Fdrussac for June, 1830 (vol. xiii. p. 438), and reprinted in Liouville's Journal, vol. xi. p. 398. An account of Galois's method (completed and extended in some respects) will be found in M. Serret's Cours d'Algebre Sup^rieure, lejon 25. The theory has also been independently investigated by M. Schonemann, who seems to have been unacquainted with the earlier researches of Galois (see Crelle's Journal, vol. xxxi. p. 269, and vol. xxxii. p. 93). In several of Cauchy's arithmetical memoirs (see in particular Exercices de Mathematiques, vol. i. p. 160, vol. iv. p. 217 ; Comptes Kendus, vol. xxiv. p. 1117 ; Exercices d'Analyse at de Physique Math^matique, vol. iv. p. 87) we find observations and theorems relating to it. Lastly, in a memoir in Crelle's Journal (vol. liv. p. 1) M. Dede- kind has given (with important accessions) an excellent and lucid resume of the results obtained by his predecessors. In the following account of the principles of this theory, the functional symbols F, cp, \|/-, ... will represent (as in general throughout this Report) rational and integral functions having integral coefiicients ; we shall use ji) to denote a prime modulus, and x an absolutely indeterminate quantity. As we shaU have to consider the functions F {x), f{x), -^ (x), etc., only in relation to the modulus p, we shall consider two functions F^ {x) and F.^ (x), which differ only by multiples of p, as identical, and we shall represent their identity by the congruence Fi{x) = F2{x), mod p, which is equivalent to an identical equation of the form F^ (x) = Ffi{x)+p(p (x). The designation ' modular function,' * See Disq. Arith., Arts. 11 and 43. + Galois was bom October 2G, 1811, and lost his life in a duel, May 30, 1832. He was consequently eighteen at the time of the publication of the note referred to in the text. His mathematical works are collected in Liouville's Journal, vol. xi. p. 381. Obscure and fragmentary as some of these papers are, they nevertheless evince an extraordinary genius, unparalleled, perhaps, for its early maturity, except by that of Pascal. It is impossible to read without emotion the letter in which, on the day before his death and in anticipation of it, Galois endeavours to rescue from oblivion the unfinished researches which have given him a place for ever in the history of mathe- matical science. 150 REPORT ON THE THEORY OF NUKBER8. [Art. 69. which has been introduced by Cauchy (Comptes Rendus, vol. xxiv. p. 1118), will serve (though, perhaps, not in itself very appropriate) to indicate that the function to which it is applied is thus considered in relation to a prime modulus. Since in any modular function we may omit those terms the co- efficients of which are multiples of p, we shall always suppose that the coefficient of t le higri e.st power of ic in the function is prime to p. If F{x)=fi(x)xfi{x), mod p, fi{x) and ^(x) are each of them said to be divisors of F{x) foi- the modulus p, or, more briefly, modular divisors of F{x), or even simply divisors of F{x) when no ambiguity can arise from this elliptical mode of expression. If a be a function of order zero, i. e. an integral number prime to ^, a is a divisor, for the modulus p, of every other modular function; so that we may consider the p — 1 terms Oj, a^, a^, ....Op.i, of a system of residues prime to p, as the units of this theory, and, in any set of j9 — 1 associated functions a^Fix), a^F(x), ..., a^_^F{x), we may distinguish that one as primary in which the highest coefficient is congruous to unity (mod x>)- If F{x) be a function which is divisible (mod p) by no other function (except the units and its own associates), F{x) is said to be a prime or irre- ducible function for the modulus p. And it is a fundamental proposition in this theory, that every modular function can be expressed in one way, and one way only, as the product of a unit by the powers of primary irreducible modular functions. The demonstration of this theorem depends (precisely as in the case of ordinary integral numbers) on Euclid's process for finding the greatest common divisor, which, it is easy to show, is appUcable to the modular functions we are considering here. For, if ^pxix) and (^^{x) be two such functions [the degree of ^2 (*) being not higher than that of ^1 (x)], we can always form the series of congruences ^1 {x) = qi (x) (j>2 (x) + Ti 03 (x), mod p, <l>2 (x) = qi (x) 03 (x) + r^ 04 (x), mod p, in which r,, r^, ... denote integral numbers, qi{x), q^ix), ... modular functions, and 03 (x), 04 (x), . . . primary modular functions, the orders of which are suc- cessively lower and lower, until we arrive at a congruence <i>ic {x) = qu (x) 0i + 1 (x) + rj 0i + 2 (x), mod p, in which rj=0, modp. The function 0i,+i(x) is then the greatest common Art. 69.] REPORT ON THE THEORY OF NUMBERS. 151 divisor (mod p) of the given functions ^i {x) and ^^ (x) ; and, in particular, if <pj^^i{x) be of order zero, those two functions are relatively prime. We may add that, if ^ be the Resultant of (pi (x) and <^2 (*)) the necessary and sufficient condition that these functions should have a common modular divisor of an order higher than zero is contained in the congruence R = 0, mod p ''"' — a theorem exactly corresponding to an important algebraical proposition. From the nature of the process by which the greatest common divisor is determined, we may infer the fundamental proposition enunciated above, by precisely the same reasoning which establishes the corresponding theorem in common arithmetic. Similarly, we may obtain the solution of the following useful problem : — 'Given two relatively prime modular functions A„^ and A,i, of the orders m and n, to find two other functions, of the orders m — 1 arid n — 1 respectively, which satisfy the congruence ^m -^n-i - A -3r„_i = 1, mod ^.' The assertion, that f{x) is a divisor of F{x) for the modulus p, is for brevity expressed by the congruential formula i^{x) = 0,mod[>,/(x)], which represents an equation of the form F{x)=p<p{x)+f{x)>\r{x). Similarly the congruence F,{x) = F,{x),modi[p,f{x)l is equivalent to the equation F,{x) = F,{x)+p<i>{x)+f{x)>\.{x). If f{x) be a function of order m, it is evident that any given function is congruous, for the compound modulus [p, f{xj\ to one, and one only, of the p^ functions contained in the formula ao + <*i*'+ ••• +«m-i''^~S ii^ which «o, Oi. •••, «m-i "lay have any values from zero to p — 1 inclusive. These p'^ functions, therefore, represent a complete system of residues for the modulus A congruence F{X) = 0, mod [j9, /(a;)], is said to be solved when a functional value is assigned to X which renders the left-hand member divisible * See Canchy, Exercices de Mathematiques, vol. i. p. 160, or M. Libri, M6moires de Math6- matique et de Physique, pp. 73, 74. But a proof of this proposition is really contained in Lagrange's Additions to Euler'a Algebra (sect. 4). 152 KEPORT ON THE THEORY OF KUMBERS. [Art. 70. by f{x) for the modulus p ; and the number of solutions of the congruence is the number of functional values (incongruous mod [i>,/(a!)]) which may be attributed to X. The coefficients of the powers of X in the function F {X) may be integral numbers or fiinctions of x. The linear congruence AX=B, mod [p,f{x)l in which A and B denote two modular functions, is, in particular, always resoluble when A is prime to /(x), mod p, and admits, in that ctuse, of only one solution. We shall now suppose that the function f{x) in the compound modulus \.P'f{^y\ ^ irreducible for the modulus p, — a supposition which involves the consequence that, if a product of two factors be congruous to zero for the modulus \_p,f{x)], one, at least, of those factors is separately congruous to zero for the same modulus. We thus obtain the principle (cf. Art. 11) that no congruence can have more solutions, for an irreducible compound modulus, than it has dimensions. For, if Jl'=^, mod [p,f{xy\, satisfy the congruence F^{X) = 0,mod[p,f{x)l we find F,„{X) = F„,{X)-F„,{^) = iX-0 F,,_,{X), mod [p,f{x)l F„,_i(X) denoting a new function of order m — 1; whence it follows that if the principle be true for a congruence of m — 1 dimensions, it is also true for one of m dimensions ; i. e. it is true universally. 70. Extension of Fermafs Theorem. — Let 6 denote any one of the ^^ — 1 residues of the modulus [p, /(a?)] which are prime to /(«)] ; it may be shown, by a proof exactly similar to Dirichlet's proof of Format's Theorem, that e^'^-^ =1, mod [p, fix)] (A) This result, which is evidently an extension of Fermat's theorem, involves several important consequences. It implies, in the first place, the existence of a theory of residues of powers of modular functions, with respect to a compound modulus, precisely similar to the theory of the residues of the powers of integral numbers with regard to a common prime modulus. A single example (taken from M. Dedekind's memoir) will suffice to show the exact correspondence of the two theories. The modular function is or is not a quadratic residue of f{x), for the modulus p, according as it is or is not possible to satisfy the quadratic con- gruence A'^^=0, mod [j9,/(a;)]. In the former case satisfies the congruence g . (pm_i) _ j^ ^^ |-^^ y^^^-j . .^ ^^^ ^^^^^ qHp^-d = _ 1^ jnod [p, f{x)]. And, further, if Oi and d^ be two primary irreducible modular functions of the orders Art. 70.] KEPOET OK THE THEORY OF NUMBERS. 153 m and n respectively, and if we use the symbols f— 1 and [—1 to denote the positive or negative units which satisfy the congruences 0^^(P--i) = (^), mod {p, e,), and 0,^(i"»-i) = (|), mod {p, e,), respectively, these two symbols are connected by the law of reciprocity 6,- m-^-'^'-m But the equation (A) admits also of an immediate application to the theory of ordinary congruences with a simple prime modulus. In that equation let us assign to 6 the particular value x ; we conclude that the function x^'"-'^-\ is divisible for the modulus p \>j f{x), i.e. by every irreducible modular function of order m. Further, if d be a divisor of m, x"""'"'-! is algebraically divisible hj x'^^-'^-l ; whence it appears that aj"*""' — 1 is divisible, for the modulus p, by every function of which the order is a divisor of m. But it is easily shown that a;*''"-i - 1 is not divisible (mod pi) by any other modular function, and that it cannot contain any multiple modular factors. Hence we have the indeterminate congruence a;i"--i-l = n/(a;), modj9, (B) in which f{x) denotes any primary and irreducible function, the order of which is a divisor of m, and the sign of multiplication IT extends to every value of f{x). This theorem, again, is a generalisation of Lagrange's inde- terminate congruence (Art. 10). We may infer from it that, when m is>l, the number of primary functions of order m, which are irreducible for the modidus p, is ■| _ TO ?* TO — jp" - 2^9i + "Lp^- S^vJ^H- . . , 1 , qi, q^, ... denoting the different prime divisors of m. As this expression is always different from zero, it foUows that there exist functions of any given order, which are irreducible for the modulus p. A congruence F {x) = 0, mod p, may be considered resolved when we have expressed its left-hand member as a product of irreducible modular factors. The linear factors (if any) then give the real solutions ; the factors of higher orders may be supposed to represent imaginary solutions. We have already observed that even when all the modular factors of F{x) are linear, we possess no general and direct method by which they can be assigned ; it is hardly 164 REPORT ON THE THEORY OF NUMBERS. [Art. 71. necessary to add that the problem of the direct determination of modular factors of higher orders than the first, presents even greater difficulties. Nevertheless the congruence (B) enables us to advance one step toward the decomposition of F{x) into its irreducible factors; for, by means of it, we can separate those divisors of F{x) which are of the same order, not, indeed, from one another, but from all its other divisors. We may first of all suppose that F{x) is cleared of its multiple factors, which may be done, as in algebra, by investigating the greatest common divisor of F{x) and F'{x) for the modulus p. The greatest common divisor (modp) of F{x) and x''~^ — l will then give us the product of all the linear modular factors of F{x); let F{x) be divided (mod p) by that product, and let the quotient be F^ (x) ; the greatest common divisor (mod p) of Ft^{x) and x'^*~^ — l wUl give us the product of the irreducible quadratic factors of F{x); and by continuing this process, we shall obtain the partial resolution of F{x) to which we have referred. 71. Imaginary Solutions of a Congruence. — We have said that the non- linear modular factors of F(x) = 0, mod p, may be considered to represent imaginary solutions. These imaginary solutions can be actually exhibited, if we allow ourselves to assign to x certain complex values. The following proposition, which shows in what manner this may be effected, is due to Galois : — ' If f{x) represent an irreducible modular function of order in, the con- ^"^'" F(e)=0,mod[pJix)l is completely resoluble when F{x) is an irreducible modular function of order m, or of any order the index of which is a divisor of m.' To establish this theorem, write for a; in equation (B) ; we find 0P'»-i_l = ni^(0), mod_p, the sign of multiplication II extending to every irreducible modular function having w or a divisor of m for the index of its order. But the congruence e^-'i^l, mod[j9,/(x)], admits of as many roots as it has dimensions ; therefore also every divisor of ep"-! — 1 (and, in particular, the function F{d) considered as a congruence for the same compound modulus) admits of as many roots as it has dimensions. {Add that no two irreducible congruences whose indices divide m can have a root in common. } Let the order of the congruence F(6)=0,mod[p,/{x)'], be S, and let Art. 72.] REPORT ON THE THEORY OF NUMBERS. 155 any one of its roots be represented by r ; it may be shown that all its roots are represented by the terms of the series r, r^, r''^, ...,r^ ~ . For, if F(r) =0,mod[p,f(x)], we have also F{rP)= [F (r)] " = 0, mod [ 23, /(a;)] , and similarly F(r''')=[F{;r)y^ = 0, mod [p,f{x)'] ; and so on ; so that r, r", r''^, ..., r**'"' are all roots of F{e) = 0,mod[p,f{x)l It remains to show that these S functions are all incongruous, mod \_p,f{x)]- If possible let r"^' ''^^ = r'>\ mod [p, f{x)] , Ic and k' being less than S ; we have, raising each side of this congruence to the power ^'~'', r^ "^ =r'' , mod [Pffi^y], i.e. r"' =r, or r*~'=l, mod [p, /(«)], observing that r^ =r, mod [p,y(x)], because r^ ~' — 1 is divisible by F (r) for the modulus p. We conclude, there- fore, that r is a root, mod \_p,f{x)], of some irreducible modular divisor of the function 0*~' — 1, i.e. of some irreducible function of an order lower than S, because k is less than ^ ; r is therefore a root, mod \_p, /{xj], of two different irreducible modular functions, which is impossible *. If, therefore, we suppose x to represent, not an indeterminate quantity, but a root of the equation f{x) = 0, we may enunciate Galois' theorem as follows : — 'Every irreducible congruence of order m is completely resoluble in complex numbers composed with roots of any congruence which is irreducible for the modulus p, and which has m or a multiple of m for the index of its order. ' And aU its roots may be expressed as the powers of any one of them.' 72. Congruences having Powers of Pnmes for their Modules. — It remains for us to advert to the theory of congruences with composite modules — a subject to which (if we except the case of binomial congruences) it would seem that the attention of arithmeticians has not been much directed. We shall suppose, first, that the modtdus is a power of a prime number. The theorem of Lagrange (Art. 11), and the more general proposition of * {Let 60 = the common divisor of k and 8, since r*" ' = 1, r*" ~' = 1, we can prove that r*^ = 1, mod [p,/(a;)], but 0*^"'— 1 = a product of irreducible functions whose indices divide to; therefore r is a root of two different irreducible functions whose indices divide m, and this is impossible.} X 2 156 REPORT ON THE THEORY OF NUMBERS. [Art. 72. Art. 69, In which it is (as we have seen) included, cannot be extended to congruences having powers of primes for their modules. Let the proposed congruence be F{x) = 0, mod p*"; and let us suppose (what is here a restriction in the generality of the problem) that the coeffi- cient of the highest power of x in F{x) is prime to p, or, which comes to the same thing, that it is unity. Let F{x) = Px QxR ... mod p, where P,Q,R,... are powers of different irreducible modular functions. It may then be shown that F{x) = P' X Q' X R'..., mod jp", where F', Qf, R, ... are functions of the same order as P, Q, R, ..., respectively congruous to them for the modulus 2>, and deducible from them by the solution of linear congruences only. We have thus the theorem that F(x), considered with respect to the modulus jj", can always be resolved in one way and in one way only, into a product of modular functions, each of which is relatively prime (for the modulus p) to all the rest, and is congruous (for the same modulus j)) to a power of an irre- ducible function. We may therefore replace the congruence F (x) = 0, mod p^, by the congruences P' = 0, mod p^, ^=0, modjp", R = Q, mod^™, But no general investigation appears to have been given of the peculiarities that may be presented by a congruence of the form P'=0, modp^, in the case in which P is a power of an irreducible function (mod p), and not itself such a function — a supposition which implies that the discriminant of F{x) is divisible by p. If, however, P be itself an irreducible function, the congruence P' = 0, raodp", gives us one and only one solution of the given congruence if P be linear ; or, if P be not linear, it may be considered as representing as many imaginary solutions as it has dunensions. In particular, if we consider the case in which all the divisors P, Q, R, ... are linear, we obtain the theorem : — ' Every congruence which, considered with respect to the modulus p, has as many incongruous solutions as it has dimensions, is also completely resoluble for the modulus ^*" ; and has as many roots as it has dimensions, and no more.' If a;=ai, mod ^, be a solution of the congruence F{x) = Q,mod p, and if that congruence have no other root congruous to o^, the corresponding solution x = a„, mod^", of the congruence P(x) = 0, modp", may be obtained by the solution of linear congruences only — a proposition which is included in a preceding and more general observation. The process is as follows : — If, in the equation F{a, -H kp) ^F{<h) + kp F' (aO -h ^' F" (a,) + . . ., Art. 73.] REPORT ON THE THEORY OF NUMBERS, 157 we determine h by the congruence - i^(ai) + hF' (a,) = 0, mod p, Jr (which is always possible, because the hypothesis that (« — a^)^ Is not a divisor of F{x), mod p, implies that F' {a^ is not divisible by p*), and then put a2 = ai + kp, mod p^, we have F{a.^ = 0, mod^^. Similarly, from the expansion F {a., + kp-) = F {a,) + hp^ F' (a,) + . . ., a value of k may be deduced which satisfies the congruence i'^(a2 + ^j>^) = 0, or F{a;^ = 0, mod p^ ; and so on continually until we arrive at a congruence of the form F {a„) = 0, mod p"*. But when F («) is divisible (for the modulus p) by {x — ay or a higher power of x-a, the congruence F (x) = 0, mod p"*, is either irresoluble, or has a plurality of roots incongruous for the modulus p™ but all congruous to a for the modulus p. Thus the congruence {x — ay + kp{x — b) = 0, mod p^, IS irresoluble, unless a=b, mod p ; whereas if that condition be satisfied, it admits oi p incongruous solutions, comprised in the formula x=a + /ip, mod p^, m- = 0, 1, 2, 3, ..., p — 1. 73. Binomial Congruences having a Power of a Prime for their Modulus. — If M be any number, and -^ [M) represent the number of terms in a system of residues prime to M, it will follow (from a principle to which we have ah-eady frequently referred : see Arts. 10, 26, 53, 70) that every residue of that system satisfies the congruence a;i''<^'=l, mod M, — a proposition which is well known as Euler's generalisation of Fermat's theorem f. In particular, when M=p", we have a;'""^'''"'"' = 1, mod p". This congruence has, consequently, precisely as many roots as it has dimensions — a property which is also pos- sessed by every congruence of the form x^=\, mod^:**", d denoting a divisor of (p — l)i>"~^ This has been established by Gauss in the 3rd section of the Disqulsitiones Arithmeticae, by a particular and somewhat tedious method \. The simpler and more general demonstration which he intended to give in the 8th section J, was perhaps in principle Identical with the following; we exclude the case j? = 2, to which indeed the theorem itself is inapplicable : — • If jP'(a;) = {x—O'^ <f> {'')> inod p, where tf) (Oj) is not divisible by p, we have F' {x) = <p{x) + {x—a^)<l>'{x),modp, or F' (x) = </> {a^), mod p. t Euler, Comment. Arith. vol. i. p. 284. i Disquisitiones Arithmeticse, Arts. 84-88. See also Poinsot, Reflexions sur la Th^orie des Nombres, cap. iv. Art. 6. § Disquisitiones Arithmeticse, Art. 84. 158 REPORT ON THE THEORY OF NUMBERS. [Art. 74. Let d = Sp", S representing a divisor of p-1, and n being Sm-l; and let lis form the indeterminate congruence £c'-l = (a;-ai) (cc-aj) ... (x-aj), mod^""", which is always possible, because a^ - 1 = 0, mod p, has S incongruous roots. It is readily seen that, if A and B represent two numbers prime to p, and if A = B, mod p", A'^=B'^, mod p'-*-'; and conversely, if A'^=B^, mod p"-*-; A = B, mod p^* By applying this principle it may be shown that x»p" - 1 = {x"" - <") (a;"" - a/") . . . {x^" -. a/"), mod p". For if we divide x**"-! by jc'^-ai"", the remainder is V*"-!. But, because ai«=l, mod p"-", ai»^"=l, mod p" ; i.e. xi^'-ai"" divides x**""-! for the modulus p^. Similarly x*"" - 1 is divisible (mod p^) by x"" - Oa^*", etc. ; and since all these divisors are relatively prime for the modulus p, x**^ — 1 is divisible (modjp") by their product; i.e., a?p^-l = (cc"" - Oi^") (xi-" - a/") . . . (x"" - a/"), mod p". We have thus effected the resolution of x*^" - 1 into factors relatively prime, each of which is congruous (mod p) to a power of an irreducible function ; since evidently (x"" - «*■") = (x — a)*", mod p. To investigate the solutions of x'*"-l = 0, mod_p» we have therefore only to consider separately the S congruences included in the formula xJ'" = a^", mod p". But each of these congruences (by virtue of the principle already referred to) admits precisely j)" solutions, viz. the p" numbers (incongruous, mod p"') which are congruous to a, mod p"~". The whole number of solutions of x^i"" -1 = 0, mod p"*, is therefore equal to the index 5^" of the congruence. It further appears that the complete solution of the binomial congruence x*?" — 1 = 0, may be obtained by a direct method, when the complete solution of the simpler congruence x* — 1 = 0, mod p>, has been found. For we may first (by the method given in the last article) deduce the complete solution of x' — 1 = 0, modj9™"", from that of 0:^-1 = 0, modj?; and then the roots of a?^^ — 1 = 0, mod p^, can be written down at once. 74. Primitive Roots of the Powers of a Prime. — AU the elementary pro- * li A=B, mod p'', but not, mod p**^, we have A —B-\-kp^, where k is prime to p. Hence ii** = (5 + hpY = JS*^ + kB^-'^p'*'- + Kp'-^\ K denoting a coefficient divisible by p ; or A'^ = B^, mfAp''^", but not, yanAp*^"^^; because kB'^~^ is prime to p. This result implies the principle enunciated in the text. Art. 76.] EEPORT ON THE THEORY OF NUMBERS. 159 perties of the residues of powers, considered with regard to a modulus which is a power of a prime number, may be deduced from the theorem just proved. In particular, the demonstration of the existence and number of primitive roots (Art. 12) is applicable here also ; so that we have the theorem : — 'There are p'""^ (jp — 1) V' (i^- 1) residues prime to ^y", the successive powers of any one of which represent all residues prime to p™.' These residues are of course the primitive roots of p^. If 7 be a primitive root of p, of the p numbers included in the formula y-\-kp (mod p"-), p — 1 precisely wiU be primitive roots of p''. For y + kp is a primitive root of ^- unless {y + kpy ~^ = 1, mod p'^ ; and the congi-uence a;P-i = l^ mod^)^, has always one, and only one, root congruous to y for the modulus p. But every primitive root of j^^ is a primitive root of p^, and of every liigher power of p, as may be shown by an application of the prin- ciple proved in a note to the last article, or, again, by observing that every primitive root of ^ ■*■ ' is necessarily congruous, for the modulus ^:>™, to some primitive root of p", and that there are p times as many primitive roots of pm + i as of p". (See Jacobi's Canon Arithmeticus, Introduction, p. xxxiii ; also a problem proposed by Abel in CreUe's Journal, vol. iii. p. 12, with Jacobi's answer, ibid. p. 211.) 75. Case when the Modulus is a Power of 2. — The powers of the even prime 2 are excepted from the demonstrations of the two last articles — in fact, if m^S, 2" has no primitive roots. Gauss, however, has shown (Disq. Arith., Arts. 90, 91) that the successive powers of any number of the form Sn + 3 represent, for the modulus 2"*, aU numbers of either of the forms 8ft + 3 or 8n + l ; similarly all numbers of the forms 8 n + 5 and 8 ?i + 1 are repre- sented by successive powers of any number of the form 8h + 5. If, there- fore, we denote by y any number of either of the two forms 8n + 3 or 8n + 5, we may represent all uneven numbers less than 2'" by the formula ( — 1)"7^, in which a is to receive the values and 1, and /8 the values 1, 2, 3, ..., 2'"-^. A double system of indices may thus be used to replace the simple system supplied by a primitive root when such roots exist. Tables of indices for the powers of 2, and of uneven primes inferior to 1000, have been appended by Jacobi to his Canon Arithmeticus. 76. Composite Modules. — No general theory has been given of the repre- sentation of rational and integral functions of an indeterminate quantity as products of modular functions with regard to a composite modulus divisible by more than one prime. And it is possible that no advantage would be 1 60 BEPORT ON THE THEORY OF NUMBERS. [Art. 77. gained by considering the theory of congruences with composite modules from this general point of view. A few isolated theorems relating to particular cases have, however, been given by Cauchy (Comptes Rendus, vol. xxv. p. 36, 1847). Of these the following may serve as a specimen : — 'If the congruence i^(a5) = 0, mod M, admit as many roots as it has dimensions, and if, besides, the differences of these roots be all relatively prime to M, we have the indeterminate congruence F{x) = k{x- ri) [x - r^ {x - r^ ...{x- r„), mod M, h denoting the coefficient of the highest power of x in F{x).' But if, instead of considering the modular decomposition of the function F(x), we confine ourselves to the determination of the real solutions of the congruence F(x) = 0, mod M, it is always sufficient to consider the congruences F{x) = 0, mod A, F{x) = 0, mod B, F{x) = 0, mod C, etc., . . . (G) where AxBx C ...=M, and A, B, C, ... denote powers of different primes. For i£ x=a, mod A, x=b, mod B, x = c, mod C, denote any solutions of the first, second, third, ... of those congruences respectively, it is evident that, if X be a number satisfying the congruences X=a, mod A, X=h, mod B, X=c, mod C (and such a number can always be assigned), we shall have F(X) = for each of the modules A, B, C, ... separately, and therefore for the modulus M; and further, if the congruences (G) admit respectively a, j9, 7, ... incongruous solutions, the congruence F(x) = 0, mod M, will admit ax/Sx-yx... in all; for we can combine any solution of F{x)=0, mod A, with any solution of F(x) = 0, mod B, and so on*. 77. Binomial Congruences with Composite Modules. — The investigation of the real solutions of binomial congruences depends (in the manner just stated) on the investigation of the real solutions of similar congruences the modules of which are the powers of primes. With regard to the relations by wliich these real solutions are connected with one another, little of unportance has * 'Infra \i.e. in the 8th section] congruentias quascumque secundum modulom e pluribus priniis compofitum, ad congnientias quarum raoduhis est primus aut primi potestas reducere, fusius docebi- mus' (Disq. Arith., Art. 92). It is difficult to see why Gauss should have employed the word 'fusius' if his investigations extended no further than the elementary observations referred to in the text. Nevertheless it is remarkable that Gauss in the 3rd section of the Disq. Arith. sometimes speaks of demonstrations as obscure, which are of extreme simplicity when compared with one in the 4th and several in the 5th section (see in particular Arts. 53, 55, 56). Art. 78.] EEPOET ON THE THEORY OF NUMBERS. 161 been added to the few observations on this subject in the Disquisitiones Arithmeticse (Art. 92). If the modulus M be = js" q^r'..., where p,q,r,... repre- sent different primes, the congruence x*^'''^ = l, mod if, possesses no primitive roots ; for if n be the least common multiple of _p«-i(_p-l), g^-i(2-l), r«-i(r-l), ..., n will be less than, and a divisor of, ^(M). But evidently, if x be any residue prime to M, the congruence x" — 1 = will be satisfied separately for the modules p", ^, r", ..., and therefore for the modulus M; i.e., no residue exists, the first -ir {M) powers of which are incongruous, mod M. If, however, M-lp^ this conclusion does not hold, since the least' common multiple of \|/^(2) and ■^(jf^) is \/' (2j9'") itself; and we find accordingly that every uneven primitive root of p" is a primitive root of 2^"*. When, as is sometimes the case, it is convenient to employ indices to designate the residues prime to a given composite modulus, we must employ (as in the case of a power of 2) a system of multiple indices. To take the most general case, let Jf =2*|)'»g'^r<'... ; let u be any number of either of the forms Sn + S or 8w + 5, and P, Q, R, ... primitive roots of ^^ <f, r^, ... respectively. Then, if « be any given number prime to M, it will always be possible to find a set of integral numbers e„, ft)„, a„, /3„, 7„, ... satisfying the conditions ( - 1)'- u"' = n, mod 2« ; S f„ < 2, ^ w„ < 2^-^, 0^a„<_p«-J(jp-l), 0<7„< r"-i(r-l); and these numbers form a system of indices by which the residue of n for each of the modules 2*, p", <f, ?*, ... (and consequently for the modulus M) is completely determined. (See Dirichlet's memoir on the Arithmetical Pro- gression, sect. 7, in the Berlin Memoirs for 1837.) 78. Primitive Roots of the Powers of Complex Primes. — Dirichlet has shown * that, in the theory of complex numbers of the form a + hi, the powers of primes of the second species (see Art. 25) have primitive roots ; in fact, if a + hi be such a prime, and N (a + hi) = a'^ + h'^=^p, every primitive root of p'^ is a primitive root of (a -f hif. On the other hand, if g be a real prime of the form 4»-i-3, q^ has no primitive roots in the complex theory. For in general, if ilf be any complex modulus, and M=a'^h^c'^...,a,h, c, ... being P"" = n, mod p" ; Q^' =. n, mod g* ; R""' = n, mod r" ; * See sect. 2 of the memoir, Untersuchungen iiber die Theorie der complexen Zahlen, in the Berlin Memoirs for 1841. 162 REPORT ON THE THEORY OF NUMBERS. [Art. 78. different complex primes, and if A=N{a), B = N(h), C=N{c), etc., the number of terms in a system of residues prime to Jf , is A'-'iA-l) B^-^B-l) Cy-'{C-1) ...; and if we denote this number by ^(M), every residue prime to M will satisfy the congruence a;*<"' = l, mod M, which here corresponds to Euler's extension of Fermat's Theorem. If If =5"*, this congruence becomes ^q^^'^-'^Hq^-i) = j^ mod 5" ; but it is easily shown that every residue prime to 5"* satisfies the congruence ««""'<«'-!) = 1, mod 5"; i.e., q^ has no primitive roots, because the exponent q'^-^ (q^—l) is a divisor of, and less than, q^^'"~^''{q^-l). Nevertheless two numbers, y and y, can always be assigned, of which one appertains to the exponent q"~^(q^ — l) and the other to the exponent q^~^, and which are such that no power of either of them can become congruous to a power of the other, mod 5", without becoming congruous to unity ; from which it will appear that every residue prime to g" may be represented by the formula y" y", if we give to x all values from to (g2_l)^«»-'_l inclusive, and to y all values from to ^'-'-1 inclusive. The corresponding investigations for other complex numbers besides those of the form a + bi have not been given. We here conclude our account of the Theory of Congruences. The further continuation of this Report will be occupied with the Theories of Quadratic and other Homogeneous Forms, [The Additions to Arts. 16, 20, 22, 24, 25, 36, 37, and 38 to Part I of the Report (1859), inserted in their proper places, see footnote p. 57, were published at the end of this Part of the Report.] VIL REPORT 01^ THE THEORY OF NUMBERS. Paet III. [Report of the British Association for 1861, pp. 292-340.] (B) Theory of Homogeneous Forms. 79. ±ROBLEM of the Representation of Numbers. — A rational and integral homogeneous function (a quantic according to the nomenclature introduced by Mr. Cayley), of which the coefficients are integral numbers, is, in the Theory of Numbers, termed a form (Disq. Arith., Art. 266). The form is linear, quad- ratic, cubic, biquadratic or quartic, quintic, &c., according to its order in respect of the indeterminates it contains ; and binary, ternary, quaternary, &c., according to the number of its indeterminates. Thus x'^ + y^ is a binary quadratic form, a5* + 2/' + z* — 3ic?/z a ternary cubic form. A form is considered to be given, when its coefficients are given numbers ; and a number is said to be represented by a given form, when integral values are assigned to the indeterminates of the form, such that the form acquires the value of the number. If the values of the indeterminates are relatively prime, the representation is said to be primi- tive ; if they admit any common divisor beside unity, it is a derived repre- sentation. Thus 13 and 8 can be represented hy x^ + y"^; for 3^-}- 2^ = 13, 22 -f 2^ = 8 ; and the first of these representations is primitive, the second is derived. The first general problem, then, that presents itself in this part of the Theory of Numbers, is the following, ' To find whether a given number is or is not capable of representation by a given form, and, if it is, to find all its representations by that form.' The number of different representations of a given number by a given form may be either finite or infinite ; in the former case the complete solution of the problem of representation consists in y 2 164 REPORT ON THE THEORY OF NUMBERS. [Art. 80. the actual exhibition of the different sets of values that can be given to the indeterminates of the form : in the latter case it consists in assigning general formulae, in which all those values are comprised. It is in either case sufficient to consider primitive representations only ; for if the given form / be of order m, and the given number N be divisible by the m*^ powers di'", d^, ..., the derived representations of N by / coincide with the primitive representations of T— , -T— , . . . by the same form, c?," d.p ^ 80. Problems of the Transformation and Equivalence of Forms. — A form fipifi, x\, ...,x'n) is said to be contained in another form/i(xi, ajg, ...,x„), when f arises from/, by a linear transformation of the type Xi^ OiiX I'vCl^^sX 2T •^,n'^ n) X^ = ^,1'^ I T 6^2,2''' 2' '^.n"^ n> •*'» ~ '''■n,l**'l"l"^n,2* 2 1" "i^n.ti'^ n> in which the coefficients a,_j are integral numbers and the determinant is dif- ferent from zero *. This transformation we may, for brevity, describe as the transformation \a\. When la| is a unit-transformation, i.e. when the deter- minant of \a\ is a positive or negative unit, the inverse transformation of \a\, which will be a transfoi-mation of the same type as | a | , will have all its co- efficients integral numbers; so that in this case f, which contains^, is also contained in it. When each of two forms is thus contained in the other, they are said to be equivalent. Ifyj contain^, and _^ contain ^,/i will contain y^; for if /, be changed into f by the transformation \a\, and f into f by the transformation | & |, it is clear that^j will be changed into^^ by a transformation \T\, of which the constituents are defined by the equation The transformation | T' | is said to be compounded of the transfonnations | a \ and 1 6 1, and this composition is expressed by the symbolic equation |ri = la|xI6|, in which it is to be observed that the order of the symbols | a \ and 1 6 ] is not, in general, convertible. When, in particular, f is equivalent to f, and ^ to * Gauss Bays that f^ is contained in /,, even when the determinant of transformation is zero (Disq. Arith., Art. 215). But we shall find it more convenient to retain the restriction specified in the text. Art. 80.] REPOKT ON THE THEORY OF NUMBERS. 165 f^, /i is equivalent to f^ ; i.e. forms which are equivalent to the same form are equivalent to one another. All the forms, therefore, which are equivalent to one and the same form, may be considered as forming a class. All the in- variants of any two equivalent forms have the same values ; but it is not true, conversely, that two forms which have the same invariants are necessarily equivalent. Nevertheless it may be conjectured that all forms of the same sort (i.e. of the same degree, and the same number of indeterminates), the invariants of which have the same values, distribute themselves into a finite number of classes ; and this conjectural proposition is certainly true for binary forms of all orders, and for quadratic forms of any number of indeterminates. It is readily seen that if a number be capable of representation by one of two equi- valent forms, it is also capable of representation by the other ; and that the number of representations is either finite for both, or infinite for both, and, if finite, is the same for each. The general problem, therefore, of the representa- tion of numbers (which we have already enunciated) suggests naturally the following, which we may term that of the equivalence of forms : ' Given two forms (of the same sort), of which the invariants have equal values, to find whether they are, or are not, equivalent, and if they are, to assign all the transformations of either of them into the other.' The number of transforma- tions may be either finite or infinite ; if finite, the transformations themselves, if infinite, general formulae containing them, are required for the complete solution of the problem. When fy is not equivalent to, but contains _^, the invariants of f^ are derived from those of y, by multiplication with certain powers of the modulus {i.e. of the determinant) of the transformation by which y^ is changed into f^ ; viz. if / be an invariant of yj, and if i and m be the orders of I, and of yj or mi J\, the corresponding invariant of /a is a"/, a denoting the modulus of trans- formation, and the number — being always integral. This observation enables us to enunciate with precision a problem in which the preceding is included : ' Given two forms, of which the invariants have values consistent with the supposition that one of them contains the other, to find whether this suppo- sition is true or not, and, if it is, to find all the transformations of the one form into the other.' But, in every case, the solution of the problem in this more general form may be made to depend on the solution of the problem of equivalence. For every transformation of order n, and modulus a, arises, in 166 REPOET ON THE THEORY OF NUMBERS. [Art. 81. one way and in one only, from the composition of two transformations | a \ and \v\, of which the latter is a unit- transformation, and the former one of the finite nmnber of transformations included in the formula Ml) "a,2> "^,3> > "i,n 0, ft-i, «^2,3j > "ii.n 0, 0, H^, , kin 0, (C) 0, 0, ,/*„ in which fx^xiui^y ... x fi„ = a, and < ^.•,j<m< (Phil. Trans., vol. cli. p. 312). To determine, therefore, whether the form /j can be transformed into ^ by a transformation of modulus a, we apply to /i all the transformations (C) in suc- cession, obtaining a series of transformed forms (pi, (p2> — If none of the forms <f> are equivalent to fi, fi cannot contain f^ ; but if one or more of the forms <f> be equivalent to f2,fi will contain ^, and all its transformations into f^ may be obtained as soon as the transformations of the forms (p into ^ have been determined. This is the method proposed by Gauss for binary quadratic forms (Disq. Arith., Arts. 213, 214); it is evidently of universal application; but the following modification of it possesses a certain advantage. Instead of represent- ing 1 7"! by the formula | T] = |a| x \v\, we may employ the formula | T\ = \v\ x \a\, in which 1^1 is a unit- transformation as before, and | a ] is one of the transforma- tions included in the formula (C), where, however, the inequality ^ Z;,- j- < //,• is to be replaced by < A;,- _,- < a(j' ; the transformations thus defined we shall call the transformations (C). If we now apply to f^ the inverse of each trans- formation included in (C), we shall obtain a series of forms <^i, (p^, (p^, ... of which the coefficients wiU not necessarily be integral numbers, because the coefficients of the inverse transformations are not necessarily integral. If all the forms (px,(p2,... be fractional, or if none of those which are integral be equivalent to f^, f^ cannot contain f^ ; but if some of them be integral, and equivalent toj^, it is plain thatyi contains /a, and that all the transformations of^ into_^ may be obtained by means of the transformations of ^i into those forms (p which are equivalent to it. The advantage above referred to consists in the circumstance that the rejection of the fractional forms (p diminishes the number of the problems of equivalence which must be solved to obtain the com- plete solution of the question proposed (compare Disq. Arith., Art. 284, and note.) 81. Automorphic Transfoi'mations. — The unit-transformations by which a 2,3 form passes into itself are the automorphics of the form ; thus 1, 2| is an of which Art. 81 ] REPORT ON THE THEORY OF NUMBERS. 167 automorphic of x- — 3y^. When every invariant of a form is zero, the form may pass into itself by transformations of which the modulus is different from imity; for example, x- — 4xy + 4:y\ a binary quadratic form of which the dis- I 3 2 criminant is zero, passes into itself by the transformation L ' the modulus is 4. In like manner it is to be observed that when two -forms of the same sort have all their invariants equal to zero, it may happen that each of them passes into the other by transformations of which the modulus is not a unit. But in this Report we shall have no occasion to consider these exceptional cases, whether of equivalence or of automorphism, and we shall therefore employ these terms with reference to unit-transformations exclusively. K 1 jTi I and I Tg I be automorphics of a form /, | T^ | x | T'g | and | Tg [ x 1 7\ ] are also automorphics of y"; so that, in particular, every power of an automorphic is also an automorphic. (The positive powers of a transformation are, of course, the transformations which arise from compounding it continually with itself; its negative powers are the positive powers of ite inverse. See Mr. Cayley's Memoir on the Theory of Matrices, Phil. Trans., vol. cxlviii. p. 17.) Hence, if a form have a single automorphic, of which no two powers are identical, it will have an infinite number of automorphics. The importance of auto- morphic transformations in the solution of the problems of equivalence and transformation will be apparent from the following considerations. If yi and f2 be two equivalent forms, \h\ a. given transformation ofyj into ^, |ail and I og I the general formulae representing all the automorphics otf^ and f^ respect- ively, all the transformations offi intoy^ wQl be represented by either of the formulae | oj | x ] A- 1 or ] A | x | og |. And again, if yj contain f^, and if we represent by l^il, 1^2!,... certain particular transformations ofyi into ^, obtained by compounding each transformation (C), which gives a form (f) equivalent to^, with some one transformation of (p into f^, then all the transformations of /i Intoya will be comprised in a finite number of formulae of the type |Ai|x|a2|, |A2|X|«21. 1^3|X1«2|) , joal still denoting indefinitely any automorphic of_^. Or, if we employ the second method of the preceding article, the same transformations will be represented by |a,lxlV|. |ai|x|V|, |«i'l^lV|. where | oj [ is any automorphic of yj , and | A/ 1 , 1^2'!) 1 ^s' 1 > ^^e certain particular transformations ofyj into y2, obtained in a manner sufficiently in- dicated by the method itself. It appears, therefore, that when we know all 168 REPORT ON THE THEORY OF NUMBERS. [Art. 82. the automorphics either of /i or f^, we can deduce all the transformations of/i into /a, from one of those transformations when /i is equivalent to /j, and from a certain finite number of them when /, contains, but is not equi- valent to, f-i. We may add, that when one transformation of two equivalent forms, and the automorphics of either of them are known, those of the other are known also, for we evidently have the equation |a,| = lA|-^xla,lx|A|. 82. Problem of the Representation of Forms. — We give the enunciation of one other general problem, which may be said to occupy a middle place between the problems of the representation of numbers, and of the equi- valence of forms. By using a defective substitution of the type S*! = ^^l.l*" 1 "r ^l,i^ 2> "I" ^\,n — i'^ » — r> 3^2 = Ct2,l*'' l"!" (f'2,2'^ 2'T + ^2,n — a'*' n— r> •'^n = <*n,l'*' l + '^n.a'^'z"'" + ^n.n-a"^ n-o a form^(a:i, x^, ...,x„) may be changed into another f{x\, x\, ...,x\_^ of the same order but containing fewer indeterminates. The form f is said to be rejyresented by f ; and the representation is proper or improper according as the determinants of the system do not, or do, admit of any common divisor besides unity. Our third general problem therefore is, ' Given two forms of the same order, of which the first contains more indeterminates than the second, to find whether the second can be represented (properly or improperly) by the first, and, if it can, to assign all the representations of which it is susceptible.' If the second form contain only one indeterminate {i.e. if it be an expression of the form Ax''), the problem reduces itself to that of the representation of the number A by the form f. If, again, f contains at* many indeterminates as f, the problem becomes that of the transformation of f into f. We may add that the problem of improper representation may be made to depend on that of proper representation, by methods analogous to those by which the problem of transformation depends on that of equivalence. (See Disq. Arith., Art. 284, where Gauss treats of the improper representation of binary by ternary quadratic forms.) 83. It is hardly necessary to state that what has been done towards ob- taining a complete solution of these problems is but very httle compared with what remains to be done. Our knowledge of the algebra of homogeneous forms (notwithstanding the accessions which it has received in recent times) is far Art. 84.] BEPORT ON THE THEORY OF NUMBERS. 169 too incomplete to enable us even to attempt a solution of them co-extensive with their general expression. And even if our algebra were so far advanced as to supply us with that knowledge of the invariants and other concomitants of homogeneous forms which is an essential preliminary to an investigation of their arithmetical properties, it is probable that this arithmetical investigation itself would present equal difficulties. The science, therefore, has as yet had to confine itself to the study of particular sorts of forms ; and of these (ex- cepting linear forms, and forms containing only one indeterminate) the only sort of which our knowledge can be said to have any approach to completeness are the binary quadratic forms, the first in order of simplicity, as they doubtless are in importance. Of all other sorts of forms our knowledge, to say the least, is fragmentary. We shaU arrange the researches of which we have now to speak in the following order, according to the subjects to which they refer : — 1. Binary Quadratic Forms. 2. Binary Cubic Forms. 3. Other Binary Forms. 4. Ternary Quadratic Forms. 5. Other Quadratic Forms. 6. Forms of order n decomposable into n linear factors. The theory of linear forms {i.e. of linear indeterminate equations) we shall refer to hereafter. That of forms containing only one indeterminate will not require any further notice. (1) Binary Quadratic Forms. 84. Instead of confining our attention exclusively to the most recent re- searches in the Theory of Quadratic Forms, we propose, in the following articles, to give a brief but systematic resume of the theory itself, as it appears in the Disq. Arith., introducing, in their proper places, notices, as full as our limits will admit, of the results obtained by later mathematicians. We adopt this method, partly to render the later researches themselves more easily intel- ligible, by showing their connexion with the whole theory; but partly also in the hope of facilitating to some persons the study of the Fifth Section of the Disq. Arith., which, probably owing to the obscurity of certain parts of it, is even now too much neglected by mathematicians. This section is com- posed, as Lejeime DLrichlet has observed (CreUe, vol. xix. p. 325), of two very distinct parts. The results contained in the former of the two (Arts. 153-222) 170 REPORT ON THE THEORY OF NUMBERS. [Art. 85. are for the most part those which had been already obtained by Euler, La- grange, and Legendre; but they are completed in many respects; they are derived, in part at least, from different principles, and are expressed in a terminology which has been adopted by most subsequent writers. The second part (Arts. 223-307) is occupied, after some preliminary disquisitions (Arts. 223-233), with the ulterior researches of Gauss himself We proceed then to give a summary of the definitions and theorems contained in the first of these two portions. 85. Elementary Definitions. — The quadratic form ax^ + 21x^1 + cy^ is sym- bolised by the formula (a, h, c) {x, yY, or, when it is not necessary to specify the indeterminates, by the simpler formula (a, b, c). The second coefficient is always supposed to be even ; and an expression of the form px^ + qxy + ry"^ (in which q is uneven) is not considered by Gauss as itself a quadratic form, but as the half of the quadratic form (2jp, q, 2r). The discriminant ¥-ac of the form (a, h, c) is called by Gauss the determinant of the form ; an expression which at the present time it would be neither possible nor desirable to alter. When two forms are equivalent, they are said to be properly equivalent if the modulus of transformation is -1- 1, but imprope7'ly equivalent if it is — 1. Only those forms which are properly equivalent to one another are considered to belong to the same class ; two forms which are only improperly equivalent are said to belong to opposite classes. This distinction between proper and im- proper equivalence is due to Gauss, and is of very great importance. In what follows, unless the contrary is expressly specified, we shall use the terms equi- valence and automorphism to denote proper equivalence and proper automor- phism. It is readily seen that the greatest common divisors of a, 26, c, and of a, b, c are the same for (a, b, c) and for every form equivalent to (a, b, c) ; if each of those greatest common divisors is unity, (a, 6, c) is a properly primitive form, and the class of forms equivalent to (a, b, c) a properly primitive class ; if the fiirst greatest common divisor be 2, and the second 1, the form, and the class of forms equivalent to it, are termed improperly piimitive. Every form which is not itself primitive, is a numerical multiple of some primitive form of a less determinant, and is therefore called a derived form. Thus x^-\-Zy'^iB& properly primitive form of det. -3, but 2x'^-\-2xy + 2y^ is an impro- perly primitive form of the same determinant; whUe 2x^ + 6?/^, 4a;^H-4x2/4-4«/* are derived forms of det. — 12. In all questions relating to the representation of numbers, or the equi- valence of forms, it is sufficient to consider primitive forms, as the solution of Art. 85.] REPORT ON THE THEORY OF NUMBERS. 171 these problems for derived forms is immediately deducible from their solution for primitive forms ; but in certain investigations connected with the trans- formation of forms the consideration of derived forms is indispensable. (The problem of Art. 82- coincides with that of the representation of numbers, in the case of binary forms of any order.) The nature of the quadratic form (a, h, c) depends very mainly on the value of its determinant, which we shall symbolise by D. (1) If D = 0, the form {a, b, c) reduces itself to an expression of the type m {px + qyY, p and q denoting two numbers relatively prime, and m being the greatest common divisor of a, h, c. The arithmetical theory of such expressions, which are not binary forms at all, since they are adequately represented by a formula such as mX°, is so simple, and at the same time diverges so much from that of true binary quadratic forms, that we shall not advert to it again in this Report, and in all that follows the determinant is supposed to be different from zero. (2) When Z) is a perfect positive square, the form (a, b, c) reduces itself to an expression of the type m {piX + qiy) {p^x + qzy), i.e. it becomes a product of two linear forms. Owing to this circumstance the theory of forms of a square determinant is so much simpler than that of other quadratic forms, that we shall not enter into any details with regard to them, though it is not necessary to exclude them (as is the case with forms of determinant zero) from those investigations which relate simultaneously to the two remaining kinds of quad- ratic forms ; viz. (3) those of a negative determinant, and (4) those of a positive and not square determinant. An essential difference between these two kinds of forms is, that whereas both positive and negative nvmabers can be repre- sented by any form of positive and not square determinant, forms of a negative determinant can represent either positive numbers only, or negative numbers only. For if the roots oi' a + 2b6 + cd^ = be real, it is clear that ax^ -{■2bxy + cy^ win have values of different signs, when the ratio y : x falls between the two roots and when it falls outside them ; but if the roots be imaginary, the form will always obtain values having the same sign (viz. that of a or c), whatever the ratio y : x may be. If (a, b, c) be a positive form (^. e. a form representing positive numbers only) of a negative determinant D= — A, { — a, —b, —c) is a negative form of the same determinant, and can represent negative numbers only. We see, therefore, that there are as many pos tive as negative classes for any negative determinant ; and as everything that can be said about positive forms or classes may be transferred at once, mutatis mutandis, to negative forms and classes, we shall in what follows exclude the latter from consideration, and, Z 2 172 REPORT ON THE THEORY OF NUMBERS. [Art. 86. when we are speaking of forms of a negative determinant, confine ourselves to the positive forms. Since x^ — Dy^, or (1, 0, -D), is a form of determinant D, we see that one class at least of properly primitive forms exists for every determinant ; and the class containing the form x^-Dy^ is called the principal class. Improperly primitive forms only exist for those determinants which satisfy the condition D = l, mod 4 ; since, if (a, h, c) be improperly primitive, we have h = l, mod 2, a = c = 0, mod 2. But for every determinant satisfying this condition, one class at least of improperly primitive forms exists; for (2,1, — \ is an im properly primitive form of determinant D, and the class containing it may be called the principal class of improperly primitive forms. 86. Reduction of the Problem of Mepresentation to that of Equivalence. — The problem of the representation of numbers depends, first, on the solution of a quadratic congruence, and, secondly, on the solution of a problem of equi- valence. This dependence is established by the two following theorems : — (i.) ' When the number M admits of a primitive representation by (a, h, c), the quadratic congruence x^ — D = 0, mod M, is resoluble.' For if avv' + 2bmn + cn^ = M be a primitive representation of M, let fi, v be two numbers satisfjong the equation mv — n/x = 1 ; we then find {am^ + 2bmn + cn^) {an^ + 2bfji.v + cv^) = {amii + b \mv-\-nfi\ + cnvy — D; or Q,^=D, mod M; if 12 = amti. + b \mv + n^u] + cnv. We have already referred to this result in Art. 68. The representation am^ + 2 bmn + en? of the number M by the form (a, b, c), is said by Gauss to appertain to the value Q of the congruential radical ^D, mod M. To understand this definition with precision, it is to be observed that if in the expression of Q we replace n and v by any two other numbers satis- fying the equation mv — nin = 1, the new value of Q will be of the form Q + kM; and conversely, values for m and v can always be found which shall give to amft + b[mv + nfji] + cnv any assigned value of the form Q + kM. Two different representations of M appertaining to the same value of y/D, mod M, are said to belong to the same set. (ii.) ' If M admit of a primitive representation by the form (a, b, c) apper- taining to the value of y/D, mod M, the two forms (a, b, c) and (^. °. '-^) are equivalent ; and conversely, if these two forms are equivalent, M admits Art. 86.] REPORT ON THE THEORY OF NUMBERS. 173 of a primitive representation by (a, b, c) appertaining to the value ii of ,JI), mod M.' To establish the first part of this theorem, we observe that the assertion that M admits of a primitive representation by the form (a, b, c) appertaining to the value of v^; ^od M, implies the existence of four numbers m, n, fi, v satisfying the equations inv — nfi. = l, \ am^ + 2bmn + cn^ = M, | (jt) If, therefore, we apply to {a, b, c) the transformation m, ft n, V , the resulting form will have M and Q for its first and second coefiicients respectively; its third Qt — J) coefficient will therefore be — tf — > because its determinant must be D ; ^. e. the two forms (a, b, c) and (M, Q, — ify— ) are equivalent. And, conversely, the equivalence of the two forms (a, b, c) and (^. °. ^) implies the existence of a transformation m, n n, V I of (a, b, c) into (M, Q, -^) ; i.e. it implies the existence of four numbers w, n, /n, v, satisfying the equations {k) ; or, finally, of a primitive representation of M by (a, b, c) appertaining to the value Q of ^Z>, mod M. If (A, B, C)he & form equivalent to a form (a, b, c) by which M= am^ + 2 bmn + en? is represented, and if is clear that 7,S be a transformation of (A, B, C) into (a, 6, c), it (A, B,C)(am+ jSw, ym + Sn)' = {a, b, c) (m, rCf = M. Two such representations of M by equivalent forms are called corresponding representations ; and we may enunciate the theorem, ' Corresponding repre- sentations of the same number M by equivalent forms appertain to the same value of the expression ^D, mod M ; ' the truth of this is evident from the 174 REPORT ON THE THEORY OF NUMBERS. [Art. 87. nature of the ftinction A mn + B[inv + n/i] + Cnv, which is a covariant (in respect of m, n and /x, v)toAx'^ + 2 Bxy + Cif. To obtain, therefore, all the primitive representations of a given number by a given form (a, 6, c), we investigate all the values of the expression ^D, mod M. K Qj, Qj, ... be those values, we next compare each of the forms {M, Q, -^^) with (a, 6, c). If none of them be equivalent to (a, h, c), M does not admit of primitive representation by (a, 6, c) ; but if one or more of them, as (M.O,.^A^, be equivalent to (a, h, c), let a, ^ y, S be the formula exhibiting all the trans- formations of (a, b, c) into ^ g j. (if, a„ 5-5) ; then all the primitive representations of M by (a, &, c), which appertain to the value Qi of ,y/5, mod M, are contained in the formiola (a,h,c){a,yY = M. 87. Determination of the number of Sets of Representations. — It appears from what has preceded, that if S denote a system of representative forms of determinant D (i.e. a system of forms containing one form, and only one, for every class of forms of determinant D), the number of different sets of pri- mitive representations of M by the forms of S is equal to the number of different solutions of the congruence x^^D, mod M. If, in particular, M be uneven and prime to D, it is clear that M can only be represented by properly primitive forms ; and in this case the number of solutions of the congruence x^=.D, mod M, i.e. the number of sets of primitive representations of M by the properly primitive forms contained in S, is expressed by either of the two formulae _ A /D.\ ^ /D\ in which p and S denote respectively the prime divisors of M, and those divisors of M which are divisible by no square ; while (—j and (-j\ are the quadi-atic symbols of Lagrange and Jacobi (see Arts. 16, 17, 68, 76). K /t denote the number of different primes dividing M, the common value of the two expressions n(l + (|)) and 2(f). Art. 88.] REPORT ON THE THEORY OF NUMBERS. 175 is 2'* or zero, according as the condition ( — j = 1 is satisfied by every prime divisor of M, or is not satisfied by one or more of them. When D is =1, mod 4, S will certainly contain improperly primitive forms ; and the unevenly even number 2M (where M is still supposed prime to D) will admit of primitive representation only by the improperly primitive forms contained in S (for if Q Q2_ J5 denote any root of the congruence x^=D, mod 2i¥", Q will be uneven, — ^,v- even, and the form (M, O, „ ) will be improperly primitive). And the number of sets of primitive representations of 2 if by these improperly primitive forms will be the same as the number of sets of primitive representations of M by the properly primitive forms in S. The problem of obtaining the derived representations of M by (a, h, c) depends on that of finding the primitive representations of a given number by a given form (see Art. 79). Two derived representations of M are said to belong to the same set, when the greatest common divisor of the indetermi- nates, which we will symbohse by w, is the same for each, and when the two . M primitive representations of — j, from which they are derived, appertain to the same value of ,^D, mod — . Adopting this definition, we may enunciate the theorem, ' If if be an uneven number prime to D, the whole number of sets of representations of M (and if Z) = 1, mod 4, of 2ilf) by a system of re- presentative forms of determinant i) is 2 C-r) ; d denoting any divisor of D.' We may add that, as before, M will be represented only by properly primitive forms ; and, when Z) = 1, mod 4, 2 if only by improperly primitive forms*. 88. Reduction of the Problem of Transfm'mation to that of Equivalence. — It has been shown in Art. 80, that the general problem, ' Given two forms of unequal determinants, to decide whether one of them contains the other, and if 80, to find all the transformations of the containing into the contained form,' ' The theorems of .this Article will not be found in the Disq. Arith. If, in their expression, we transform the symbols (-r-)' (-r) by the law of reciprocity, we obtain results which coincide (JL with those given by Lejeune Dirichlet in hiB memoir, ' Becherches sur Tapplicatiou etc.,' sect. 7 (Crelle, voL xsi. pp. 1-6). 176 REPORT ON THK THEORY OF NUMBERS, [Art. 89. can be reduced to the simpler problem of the equivalence of forms. For the sake of clearness we shall here point out how the first of the two general methods of that article is to be applied to quadratic forms. If of two forms / and F the former contain the latter, the determinant of i^ is a multiple of that of /by a square number, viz. by the square of the modulus of transfor-ma- tion. Let the determinant of / be D, and that of F, De^ ; also let m and n be any two conjugate divisors of e, so that mix = e. Tlien every transformation of which the modulus is e may be expressed in one way, and one only, by the formula m — 1, and m, k a, ^ 7, 5 «,|8 in which h denotes one of the numbers 0, 1, 2, 3, is any unit-transformation whatever. If, therefore, we apply to the form / all the transformations included in the formula ' 0, M (of which the number is equal to the sum of the divisors of e), we shall obtain a series of forms <Pi, <p2, ... of determinant De^. If none of these forms be equivalent to i^, -f' is certainly not contained in f; but if one or more of them, for example, f^, arising from the transformation m, h 0, At , is equivalent to F, let a, fi repre sent indefinitely any transformation of (f> into F ; then / passes into F by any one of the transformations included in the formula m, k 0,n a, ^ y, S K we take in succession for cf) every form in the series (pi, <^2> ••• which is equivalent to F, it is readily seen that the transformations off into F, which are thus obtained, are all different, and that they include all possible transformations of/ into F. We have supposed the number e to be positive, i.e. we have supposed that / contains F properly. To decide whether / contains F improperly, we have only to examine whether any of the forms cp^, (p2, ... be improperly equivalent to F ; and if any one of them be so, to combine the transformation of / into it, with its (improper) transformations into F. 89. Problem of Equivalence. — It remains to speak of the problem of equi- valence. Of the three parts of which this problem consists, viz. (1) to decide whether two given forms are equivalent or not, (2) if they are, to obtain a single transformation of one form into the other, and (3) from a single transformation to deduce all the transformations, the last only admits of being treated by a method equally applicable to forms of a positive and negative determinant. We shall therefore consider it first. The solution which Gauss has given of it Art. 89.] REPORT ON THE THEORY OF NUMBERS. 177 (Disq. Arith., Art. 162) depends on principles which are concealed (as is fre- quently the case in the Disquisitiones Arithmeticse) by the synthetical form in which he has expressed it. We shall not therefore repeat the details of his solution, but shall endeavour to point out the basis on which it rests. Let f={a, h, c) (x, yf be transformed into F={A, B, C) (x, yY by two different, but similar, transformations, «0) /3o 7o, ^0 and 7i, ^1 ; i.e. by two trans- formations of which the determinants are equal in sign as well as in magnitude to the same positive or negative number e. Let also, for brevity, Xo = aoX + ^oy, yo = 7o^ + ^oy, Xi = a^x + ^^y, Y^ = yiX + S^y, so that f{Xf,, Fo) =f(Xi, Fi) = F(x, y) ; we have then the algebraical theorem — ' The homogeneous functions F (x, y) and Xq Y^ — X^ ¥„ differ only by a munerical factor, not containing x or y.' The truth of this theorem is independent of the supposition that the coefficients of the given forms and given transformations are integral numbers. Its demonstration is implicitly contained in the formulae given by Gauss ; or it may be verified more indirectly by the consideration, that if w be a root of the equation a + 2bc6 + ca)^ = 0, we have, simultaneously, a denoting in each case the same root of the equation A + 2BQ + CQ'' = 0, an assertion which woiJd not be true, if the equal determinants ao^o~/3o7o ^^^ "i ^1 ~ ft 7i were of opposite signs. Hence the equation To + ^0 Q ^ y-^ + S^Q coincides with the equation A + 2BQ+CQ'' = 0; i.e. XoY-i-XiYf^ is identical (if we neglect a factor not containing x or y) with Fix,y). Comparing this conclusion with the identity [F{x,y)J=f{X„Y,)xf{X„Y,) 1 .^. = [aX„X, + b{X,Y, + X,Yo) + cYoY,J-I){X,Y^-X,Y,y,]- ' ' ^' we obtain a second result of the same kind — 'The function aX,X, + b(XoY, + X^Yo) + cY,Y, differs from F (x, y) only by a numerical factor not containing x or y.' Let m be the greatest common divisor of A, 2B, and C; U and T the A a 178 REPORT ON THE THEORY OF NUMBERS. [ah. 89. greatest common divisors of the coefficients of x"^, xy, and y* in X^ Y-i — Xj Y^ and a-Yo'X', + fc(XoFi + Xir„) + cFoFi respectively; m being a positive integer, but the signs of U and T being fixed by the equations F{x,y) _ X,Y,-X,Yo _ aX,X,+ h{X,Y,^X,Yo) + cY,Y^ ~^^ - U ~ T , . . . {K) wliich are implied by the two algebraical theorems that have preceded ; the numbers T, U, and m will satisfy the equation T' — DU* = m^, which is obtained by combining the equations (h) and (k), and will serve to express the relation which subsists between the transformations equations ^, -^0 Yi — Xi Y^ |8o 7o, and 7i, ^1 Solving the -^^("■y^-^f^^'-^'^i- 7i, ^1 1 = — X m 1 = — X m (C) aXoX,+ b{X,Y, + X,Y,) + cYoY, = ^F{x,y) = ^f{X„Y,), ifv liv for X^ and Y^ , we find mX^ = {T-hU)X^- cU Y„ mY,= aU Xo + {T+hU)Y^', or, finally, equating the coefficients of x and y, Ta,-U{ha, + cy,), T^,- U {h^, + cS,) Tyo+U(aa, + byo), T S,+ U{a^o+hSo) T-hU, -CU ao,^„ aU, T+bU "" y„S, If we suppose the complete solution of the indeterminate equation T'' — DU"^ = 7rC to be known, the formula (C) supplies us with a complete solu- tion of the problem, ' Given one transformation of f into F, to deduce all the similar transformations of / into F.' For if we suppose in that formula that T and U denote indefinitely any two numbers satisfying the indeterminate equation, it will appear (1) that every transformation of y into F is contained in (C) ; (2) that every transformation contained in (C) is a transformation of / into F ; (3) that no two transformations contained in (C), and corresponding to different values of T and U, are identical. Only it is to be observed that the transformations (C) are not, in general, all integral. They are so, however, when e, the modulus of transformation, is a unit, a supposition which we have not yet introduced ; i.e. when the forms f and F are either properly or im- properly equivalent ; because whence it may be inferred that a 2b and — are then evidently integral ; m m T+bU , T-bU m and m are so too. Art. 90.] REPORT ON THE THEORY OF NUMBERS. 179 90. Expression for the Automorphics of a Quadratic Form. — To find the automorphics of any quadratic form it is sufficient to consider the case of a primitive form. Putting then /'=i^, and taking for ""''"'' the identical trans- 10 . To, ^0 _ , we obtain from the formula (C) the following general expres- formation 0,1 sion for the automorphics ofy*, a, fi s 1 = — X m (D) T-bU, -cU aU,T+bU where m=l, or 2, according as / is properly or improperly primitive. The nature of this expression for the automorphics depends on the value of D. If Z) be positive and not square, let us represent the least positive numbers satisfying the equation T^ — DU'' = 7rC by 7\ and JJi ; we then have, by a known theorem, the following formula for all the solutions in which T is positive, m V m k denoting any positive or negative integral number. ). From this we can infer that if 7i. ^1 be the automorphic in the formula {D), arising from the values T^, Ui of T and U, all the other proper auto- morphics are powers of 7i, ^1 , and are included in the formula «i, Si * representing one or other of the identical transformations 1,0 0,1 and -1, 0,-1 K Z) be a negative number, the only solutions of the equation r-DU' = m' (except in two cases presently to be noticed) are T= ±m, U = 0. Hence the only proper automorphics of a form of negative determinant are the two identical transformations 1,0 0, 1 and The two excepted cases are -1, 0,-1 (1) D=-l, m = l; (2) D=-3, m = 2. In the former case we have for T and U the four values +1, 0, and 0, + 1 ; whence the proper automorphics of a form of det. — 1 are the four transforma- A a 2 180 REPORT ON THE THEORY OF NUMBERS. [Art. 91. tions supplied by the formula -h, —c a, h i If X) = — 3, m = 2, the solutions of T* + 3U' = 4: are six in all, viz. ±2, 0; +1,1; and +1, -1; whence six automorphics, comprised in the formula ia . i(l + 6) exist for an improperly primitive form of det. — 3. We may add that in each of these two cases, in addition to the proper automorphics we have found, there exist an equal number of improper automorphics. From the formula (C), compared with the theory of representation contained in Art. 86, it follows that if (a, b, c) (a, yY = Mhe any representation of M by (a, h, c), aU the representations of the same set are included in the formula r Ta-U(ba + cy) Ty+U{aa + hy) l L m ' m J ' For forms of a positive and not square determinant the number of represen- tations in each set is therefore infinite. For forms of a negative determinant the nimiber of representations in each set is, in general, two ; and if [a, 7] be one of them, the other is [ — a, —7]. But if the determinant be —1, or if the form be derived from a form of det. —1, the number of representations in each set is four ; and if the form be an improperly primitive form of det. — 3, or be derived from such a form, the number of representations in each set is six. 91. Expression for the Automorphics — Method of Lejeune Dirichlet. — We have inferred the expression (D) of the automorphics of yj from the formula (C) of which it is a particular case. But it is plain, from the general theory of Art. 81, that, when / and F are equivalent, we can conversely infer the formula (C) from (D). This method has been preferred by Lejeime Dirichlet, who obtains the automorphics of a primitive formy=(a, b, c), of which the determinant is not a positive square, by the following process (Crelle, vol. xxiv. a, /3 p. 324). If ' » t»e any rational automorphic of/, we have evidently, 7, a {ax^ + 2bxy + cy'^) = [ax + {b+ s/D) y] [ax + (6 - ^/Z)) y] , = [(aa + [& + y D] 7)0; + (a/3 + [6 + VjD] %] X [(aa+ [& - s/D]y)x + {a^ + [b-^D]S)y'], an equation which, for brevity, we may write {piX + q^y) {p2X + q^y) = {P^x+Qiy) {PiX+Q^y), Art. 91.] REPORT ON THE THEORY OF NUMBERS. 181 and which implies one or other of the two following systems (1) i>ii>.=i'xA; ^ = |; (2) i>.B = AA; ^ = |; 1 If (1) be the system which is satisfied by P2_ P. pr let ?2 Qi Pi T and U denoting rational numbers, and m stUl representing the greatest common divisor of a, 2b, c. These assumptions are legitimate, because ^ |and ^ contain no irrationality but ^/Z>, and are conjugate with regard to li/D. Substituting in the equations ^^ = ^ = -1 (T+ UVD), Pi qi m,^ " ?i = 9i = ]l(T-U^D), Pi 92 fin ^ ' for Pi,p2; Qi, qii Pi, P^; Qi,Qi, the expressions which these letters represent, and equating the rational and irrational parts, we find y, S = — X m T-bU, -cU aU,T+bU In this expression T and U satisfy the equation T^-DU^ = m^, because Pi Pi = A A- From this we infer that a^ - /37 = 1 ; fiirther, if we now introduce the condition that a, /3, y, S are to be integral and not merely rational numbers, it will follow, because y, S-a, -^ axe integral, that —U, — U, ~ U are ° m m m also integral ; i.e. that U itself, and consequently T, is integral ; so that the formula at which we have arrived coincides exactly with the formula (D). The system (2), treated in a similar manner, leads to the conclusion a^ — ^y=—l; whence it follows that that system can be satisfied by no proper automorphic of/ This method, as Dirichlet observes, has the advantage of putting in a clear light the difference between proper and improper automorphism. A proper automorphic changes each of the two factors, into which the form may be de- composed, into a multiple of itself by a complex unit of the form — \T-\- UVJD] ; 182 BEPOBT ON THE THEORY OF NUMBERS. [Art. 92. whereas improper automorphics, which only exist for particular kinds of forms, change each factor into a multiple of the other. A similar distinction subsists between proper and improper equivalence ; the radical -JD entering with the same sign, or with opposite signs, into the factors which are transformed into one another, according as the transformation is proper or improper. 92. Problem of Equivalence — Forms of a Negative Determinant. — To com- plete the solution of the problem of equivalence, we consider, first, forms of a negative, and then those of a positive and not square determinant. A form (a, h, c) of a negative determinant Z)= — A, which satisfies the conditions enunciated in the following Table, is called a reduced form. The symbols [26] etc. are used to denote the absolute values of the quantities enclosed within the brackets. Special Conditions. 1. If a = c, b^O. 2. K[26] = [a], 6^0. General Conditions. 1. [26] < [a]. 2. [26]<[c]. 3. [a]<[c]. The essential character of a reduced form is sufficiently expressed by the two symmetrical conditions [2 6]^ [a], and [26]^[c]. The third general condition (which combined with the first implies the second), and the special conditions, are, it may be said, artificial restrictions, intended to enable us to enunciate with precision the theorem that ' every class contains one, and only one, re- duced form.' To show that one reduced form always exists in any given class, we select from the given class all those forms in which the coefficient of x^ is the least ; and again, from those forms we select that one form, (a, b, c), or those two fonns, (a, b, c) and (a, —b, c), in which the coefficient of y^ is the least. The single form (a, b, c), or the two forms (a, b, c), (a, —6, c), thus obtained, will, it is easy to see, satisfy the general conditions ; and since, if a = c, or again if [2 6] = [a], the opposite fonns (a, b, c) and (a, —b, c), each of which satisfies the general conditions, are equivalent, and therefore both belong to the given class, it is clear that a form always exists satisfying the special conditions proper to these cases. That only one reduced form exists in each class may be proved by employing a principle due to Legendre (Th^orie des Nombres, vol L p. 77) : — 'lff={a,b,c) be a form satisfying the general conditions for a reduced form, /(I, 0), or a is the least number (other than zero) which can be repre- Ari. 92.] REPORT ON THE THEORY OF NUMBERS. 183 sented by/; and/(0, 1) or c is the least number which can be represented by/, with any value of the second indeterminate different from zero.' For, if we wish to find the least numbers that can be represented by / it Avill be sufficient to attribute positive values to x and y in the formula /"= ax'- — 2 hxy + cy"^, in which we suppose h positive as weU as a and c. But f{x-\,y)=f{x,y)-1l(x-y)-{a-1h)x-a{x-\), fix,y-l)=f{x,y)-2b{y-x)-{c-2b)y-c{y-\); from which equations it appears that if in the formula / (x, y) we diminish by a unit that indeterminate which is not less than the other, we diminish, or at least we do not increase, the value of / (x, y) ; a conclusion which leads immediately to the principle enunciated by Legendre. From this principle it follows that a form satisfying the general conditions of reduction is the form, or one of the two opposite forms, to which we are led by the process of selection above described. If, therefore, there be two reduced forms in the same class, they must be two opposite forms (a, h, c) and (a, — h, c). But it is easily proved that two such opposite forms, each satisfying the general conditions of reduction, cannot be equivalent, unless either [2 6] = a, or a = c; in which cases only one of the two forms satisfies the special conditions. In every case, therefore, there exists one, and only one, reduced form in each class. To obtain the reduced form equivalent to a given form, we form a series of contiguous forms, beginning with the given form and ending with the reduced form (Disq. Arith., Art. 171). Two forms of the same determinant, (a, b, c) and {a, b', c'), are said to be contiguous when c = a', and & + 6'=0, mod a'. Two contiguous forms are always equivalent; for if b+'b' = na', the former passes into the latter by the transformation ' Let, then, {a^, bg, a^) be the given form of det. — A, which is supposed not to satisfy the general conditions for a reduced form. Let &„ + ?>i = Mi «! , — ^i denoting the minimum residue of b,,, mod ai, so that [26i]<ai; and let a.^ bl + A represent the integral number a, The form (ctj, b^, a^ will be contiguous, and therefore equivalent, to {a^, b^, a^). Let a third form, {a^, b^, a^), be similarly derived from {a^, bi, a^), and let the series of contiguous forms {Ug, 6o, a,), (oi, bi, tti), (Oj, 62, a^, ... be continued until we arrive at a form 184 REPORT ON THE THEORY OF NUMBERS. [Art. 92. (a„, 5„, «„ + ,), in which a„+i^a„. We shall certainly arrive at such a form, or we should have a series of numbers a,, aj, as, ... all represented by the form (oo. ^0. <h), and yet continually decreasing for ever ; whereas a form of negative determinant can acquire only a finite number of values inferior to any given limit. The form (a„, 6„, a„+i), in which a„^a„ + i, satisfies the general con- ditions for a reduced form. For by the law of the series of forms [2fe„]<a„; and since a„<a„+i, we have also Again, the process can always be terminated in such a manner as to give a form satisfying the special conditions for a reduced form. If a„ = a„^.i, and b„ is negative, instead of stopping at the form (a„, 6„, a„), we continue the process one step further and obtain the reduced form (a„, —b„, a„). If —2b„ = a„, instead of the form (a„, &„, a„+i), we take the form (a„, — 6„, a„ + i), which is contiguous to (a„_i, &„_i, «„), for the concluding form of the series. The transformation \T„\ by which (a^, \, Oi) passes into the equivalent reduced form (a„, 6„, a„ + i), is where 0, -1 X 0, -1 X . X 0, -1 1, Ml 1, M2 1, M hi- u., = l + h . a. or if we represent the successive convergents to the continued fraction 1 . M2 1 Ms- by we may express | T, Po = 0, P. Px P. Qo' Qi' Q, -^2= — M2> ••• /*4 — • ... so that h;2 — M1M2 ■»•> •••» by the formula T 1 = p p The theory of the reduction of quadratic forms was first given by Lagrange. (See his ' Eecherches d'Arithm^tique ' in the Nouveaux Mdmoires de I'Acaddmie de Berlin for 1773 ; see also his additions to Euler's Algebra, Art. 32 ; a memoir of Euler's, 'De insigni promotione scientiae numerorum,' Opusc. Anal., vol. ii. p. 273, or Comment. Arith., vol. ii. p. 140 ; Legendre, Ttidorie des Nombres, premiere partie, sect, viii ; Disq. Arith., Arts. 171-173 ; M. Hermite in Crelle's Journal, vol. xli. p. 193.) The method is applicable to forms of a positive, Art. 93.] REPORT ON THE THEORY OF NUMBERS. 185 as well as to those of a negative determinant ; but when the determinant is positive, the reduced forms are not, in general, all non-equivalent. When the determinant is negative, it is as applicable to forms, of which the coeffi- cients are any real quantities whatever, as to those of which the coefficients are Integral numbers. We shall revert hereafter to the consequences which M. Hermite has deduced from this important observation. We have now a complete solution of the problem of equivalence for forms of a negative determinant. To decide whether two forms /i and f^ of the same negative determinant are equivalent or not, we have only to investigate the reduced forms (p^ and (p^ equivalent to /i and f^ : according as (p^ and cp^ are or are not identical, /i and /a are or are not equivalent ; and if they are equivalent, all the transformations of /i into f^ are obtained, by compounding the reducing transformation of /j, first, with the automorphics of cp^, and then with the inverse of the reducing transformation of /a. 93. Problem of Equivalence for Forms of a Positive and not Square De- terminant. — The solution of the problem of equivalence for forms of a positive and not square determinant occupies a considerable space in the Disq. Arith. (Arts. 183-196). But, as Lejeune Dirichlet has observed, in a memoir which he has devoted to this problem (' Vereinfachung der Theorie der binaren quadratischen Formen,' in the Memoirs of the Academy of Berlin for 1854, or in Liouville, New Series, vol. ii. p. 353), the demonstrations relating to it may be greatly abbreviated by employing certain known results of the theory of continued fractions. The following method does not differ materially from that proposed by Lejeime Dirichlet ; nor indeed is it, in principle, very distinct from that of Gauss, the connexion of which with the theory of continued fractions he has suppressed. We shall suppose that the forms which we consider are primitive — a sup- position which involves no loss of generality; and we shall understand, in what foUows, by a 'quadratic equation,' an equation of the form ao + 2&o^ + «i^^ = 0, in which 60^ — aoCfi is positive, and a^, ho, a^ are integral numbers without any common divisor. Such a quadratic equation we shall symbolise by the formula [fto. ^0. «i]. and we shall regard the two quadratic equations [«„, h^, a,], [ — tto, —60. — «i] as different. If >JD denote the positive square root of V — «o<*i. it is convenient to call -K-JD _^ -h+J-p and a. the first and second roots of [a,,, ho, a^ respectively; so that if we change Bb 186 REPORT ON THE THEORY OF NUMBERS. [Art. 93. the sign of the equation throughout, we change at the same time the deno- mination of the roots. Wlienever, therefore, a root of a quadratic equation, and the denomination of the root, are given, the quadratic equation itself is given. It is readily seen that if two forms (ag, hg, ch), (-^o. -^o. -^i) he pro- a,fi perly or improperly equivalent, so that 7, S transforms (a^, b^, a^) into (Ao, Bo, Ai), the corresponding roots of the quadratics a„ + 26o«4-a,«' = 0, A^ + IB^Q + A^Q!' = 0, i.e. those which are connected by the relation w = — .-^ts , are of the same, or *' o + pw of opposite, denominations, according as the equivalence is proper or improper. Let the first root of the equation [ao, \, aj be developed in a continued fraction, of which all the integral quotients are positive except the first, which has the same sign as the root. In this process we obtain a perfectly determinate series of transformed equations, each having a complete quotient of the development for its first or second root, according as it occupies an uneven or an even place in the series, counting from the proposed equation inclusive. The complete quotients eventually form a period of an even number of terms ; there exists therefore a corresponding period of transformed quadratic equations, which will be of the type [«n, 1^0, aj. [«i, A, "2], [«2. 1^2, 03], , [«2fc-i, J^aj.i, Oo]. Every equation of the period has one of its roots positive and greater than unity, the other negative and less in absolute magnitude than unity; and if we suppose (as we shall do) that we begin the period with an equation occupying an uneven place in the series of transformed equations, the positive root of any equation of the period will be its first or second root, according as it occupies an uneven or an even place in the period. To apply what has preceded to our present problem, we require the fol- lowing lemma (see sect. 2 of Dirichlet's memoir, or M. Serret in LiouvUle, voL XV. p. 153). 'K 0) and Q be two irrational quantities connected by the relation w = "^ — 3j=r, where a, j8, 7, 8 are integral and a^ — Py=±l, the developments a -f- jOii of w and Q in a continued fraction will ultimately coincide, and the same quotient wUl occupy an even or an uneven place in both aevelopments alike, if aS-0y= +1, but an even place in the one, and an uneven place in the other, if o5 — 187 = — 1. Art. 93.] REPORT ON THK THEORY OF NUMBERS. 187 A quadratic form (a^, P^, Oj) of positive determinant is said to be reduced * when the roots of [a^, jSq, ai] are of opposite signs ; the absolute value of the first root being greater, that of the second less than unity. A series of reduced forms equivalent to any proposed form ((Xo, b^, aj) can alvrays be found. For, if the first root of [cio, h^, (h\ he developed in a continued fraction, and if its period of equations (beginning with an equation occupying an uneven place in the series of transformed equations) be represented as before by [«o, ^o> «i]. [«i> A. «2]. , [oafc-i, ^2k-l, «o]> the forms (ao, jSo, a^), {a^, -fii, a^), ..., (a2t_i, - ^2k-i, «o) wUl be all reduced and all equivalent to {ag, h^, ttj). These forms, so deduced from the development of the first root of the equation [cig, b^, a^], we shall term the period of forms equivalent to (ao, &o, (h)> or, more briefly, the period of (tto, &o> *i)- It wiU be seen that each form of the period is contiguous to that which precedes it, and that the first is contiguous to the last. We can now obtain a complete solution of our problem. If (a,,, 60, a^) and (.^0, J5o, A^) are equivalent, the first roots of [(Xo, bo, aj and [^0, B^, A^l will be corresponding roots, and the developments of these two roots will ultimately coincide, giving one and the same period of complete quotients. And, p'nce the same complete quotient will occur in an even or in an uneven place alike in each development, it wUl be a root of the same denomination in the quadratic equation determining it in each development. The period of equations wUl therefore be precisely the same for each development ; and the same equation may be taken as the first equation of each period. . Con- sequently the periods of (a,,, 60, Oi), (^0, ^o> ^1) are identical. Two forms therefore are or are not equivalent, according as their periods are or are not identical. To obtain the transformations of (a,,, bo, a^) into (^0, ^o> Ai), when these two forms are equivalent, let the complete quotients in the development of the first root of [<Xo, 6o> <*i] ^6 wj, wg) •••> and let the convergent immediately o . Q preceding w„^i be — • Similarly, let Q„ + i and p^ be a complete quotient and a convergent in the development of the first root of [Ag, Bg, A^. Then, if «M ~ ^*r (where n=M, mod 2), all the transformations of (ao, 60, aj) into (^0, ^0, ^1) are comprised in the formula l^-U ?M xirix • These reduced foi-ms are not to be confounded with the reduced forms of the last article. B b 2 188 REPORT ON THE THEORY OF NUMBERS. [Art. 93. \T\ denoting any automorphic of the form corresponding to the equation of which w^ + i or Qjtf+i is a root. It should be observed that a reduced form is always a form of its own period. To prove this, we remark that reduced forms are of two kinds; they are either such as (oo, ^o, «i). where the first root of [a^, j8o, aj is positive, or such as (aj, -/Sj, a^), where the first root of [a^, -^j, oj] is negative. Now a reduced form such as (oo, $o, «i) is evidently a form of its own period, for the equation [oo, ^o, "i] is itself an equation of the period in the development of its first root. And a reduced form such as (aj, -^i, oj) is also a form of its own period.* For if we develope the second root of [oj, jSj, a^, we obtain a period of equations of which [oj, ,81, a^] is itself one. Let [oj, jSg, ag] be the equation immediately following [a,, /3i, a^] in this period; then [oj, ^j, a^ is an equation occupying an even place in the period of equations arising from the development of the first root of [a^, ^2, "s]. and consequently (a^, -,81, a^) is a form in the period of (og, ^2> "a) ; *-^- it is a form in its own period, because it is equivalent to (og, ^2, Og). It follows from this that no reduced form can be equivalent to a given form, unless it occur in the period of that form. The inequalities satisfied by the roots of any equation of a period give rise to certain Inequalities which are satisfied by its coefiicients. These in- equalities (which are not all Independent) are, (I) [« J < 2 v^Z) ; [^0] < y^ ■> [«i] < 2 v/Z) ; (ii) VD - [13,] < [aj <Vn + [^„] ; (III) ^/Z) - [/3o] < [«i] < ^/D + [^„]. The same inequalities are, of course, satisfied by the coefficients of a reduced form ; its middle coefficient is, moreover, positive. And, conversely, every form whose middle coefficient is positive and whose coefficients satisfy these inequalities is a reduced form. * {Or thus [May, 1876]: — Since (oj, — ^,, Oj) is properly equivalent to (a,, /3j, a,), the first root of [a,, — /3i, a,] gives the same period as the first root of [a„, /3„, aj, the same equations occu- pying even or uneven places in both periods alike. Hence the period of (oj, — /3,, a^ is the same as the period of (a„, ^„, a,). [July, 1876.] By developing the first root of [a,, — /3,, Oj] we do indeed obtain the period of (a,, — ^,, Oj) : but not immediately. "We ought, therefore, here to prove the theorem — that a reduced form has always one antecedent and one consequent reduced form contiguous to it. } Art. 94.] REPORT ON THE THEORY OF NUMBERS. 189 94. Improper Equivalence — Amhiguous Forms and Classes. — If it be required to find whether two forms (a, b, c) and {a', V, c') of the same positive or negative determinant are or are not improperly equivalent, it will suffice to change one of them, as (a, h, c), into its opposite (a, -h, c), and then to solve the problem of proper equivalence for {a, - h, c) and {a', h', c). If it be found that these two forms are properly equivalent, let \T\ represent any transformation of the first into the second ; then the improper transformations of (a, 6, c) into (a', &', c) will be represented by the formula 0, xin It may happen that two forms are both properly and improperly equivalent to one another ; when this is the case, each of the two forms, and every form of the class to which they belong, is improperly equivalent to itself, i.e. admits of improper automorphics. A class consisting of such forms is said to be am- biguous (classis anceps — classe amhigue). An ambiguous form is a form (a, h, c) in which 26 is divisible by a; if 2b = fia, the ambiguous form is transformed 1, A* into itself by the improper automorphic 0, -1 and if 1 T' I be the general expression of its proper automorphics, aU its improper automorphics are included by the formula ' x | ^ I • Every ambiguous form belongs to an ambiguous class, and, as we shall presently see, every ambiguous class contains ambiguous forms. To complete the theory of equivalence, we shall here briefly indicate the solution of the problem, ' To decide whether a given form is improperly equi- valent to itself or not, and if it is, to find its improper automorphics.' When the determinant is negative, it follows from the principle that two reduced forms cannot be equivalent, that no reduced form, the opposite of which is different from it and is also a reduced form, can be improperly equivalent to itself Hence the only reduced forms which have improper automorphics are those in which 6 = 0, or 2b = a, or a = c. In the two former cases the 0,1 reduced form is ambiguous, in the latter it has the improper automorphic 1,0 and is moreover contiguous and therefore equivalent to the ambiguous form (2 a — 2 6, a — b, a). These considerations supply a sufficient criterion for de- ciding whether a form of negative determinant is {improperly} equivalent to itself or not. If it is, its improper automorphics are given by the formula |r|x|T|xir|-i; |T| denoting the reducing transformation of the given form. 190 REPORT ON THE THEORY OF NUMBERS. [Art. 94. and I T I any improper automorphic of the reduced form. For forms of a positive determinant,* we observe that if be the period of (a, b, c), the period of (a, — 6, c) is («0. ~^24-l. «2l-l). («2ifc-l> ^2k-2> 02k-2)y >(«!» ^0> «o)- For (a, -&, c) is equivalent to {a„, -^2k-i> «2i-i), because (a, b, c) is equivalent to («2jt-i> ~/32t-i> «o) ; and, by a known theorem, the period of equations in the development of the second root of (a, 6, c) is [«0» "Ai-lJ <'2«;-l]» ['»2j!-l> ""P2fc-2> ^ik-i]} > [«1» ~ ^0) «0j) the equation [a,,, — ;82i_i, aaj.i] occupying an even place in the development; this period is therefore the period of equations in the development of the first root of [ao> ~ ^2k-i> «2i-i]; *-^' the period («0. ~^2k-l> <'2fc-l)> («2J;-1> ^2k-2J «2j-2)) > («1> /^Oj «o) is the period of (a„, —^2k-i> «2j;-i). or, which is the same thing, of (a, —b, c). If we now suppose that (a, 6, c) is improperly equivalent to itself, it wiU be properly equivalent to (a, —b,c); and these two forms wUl have the same period, which we shall represent by {p^, qo, Pi), {pi, qi, P2), &c. K {p^, q^, Px + i) he any form of this period, the associate of {p)^,q^,p^^.i), i. e. the form (p\ + i,qK, P\), will also be a form of the period, and the indices of these two forms in the period wUl difier by an uneven number, because the signs of the numbers i>A.> JPx + i> ••• are alternate. From this we can infer that there will be two different forms in the period, each of which will be immediately preceded by its own associate ; so that the type of the period will be {pa, qo,pi), (pu qi,p^, , (pfc-i, qk-i,pi), {Pk, qk-i,Pk-i), {pk-i, qk-2,Pk-2), , {pi, qo,Po), where for simplicity we have supposed that {po, qo, p^ is one of the two forms which is preceded by its associate ; the other is (pt, qk-i, Pk-i)- These two forms are ambiguous, for it follows from the contiguity of each form to that which precedes it, that 250 = 0, modpoi ^qk-i = ^, vaod p^. We arrive therefore at the conclusion that the period of every ambiguous class contains two ambiguous forms ; either of which enables us, as in the case of forms of a negative determinant, to obtain all the improper automorphics of any form of the class. * {Here, again, it is not necessary to recur to the definition of the period of reduced forms equi- valent to a given form; the associated period is a period of reduced forms equivalent to (a, —b, «)}. Art. 95.] REIPORT ON THE THEORY OF NUMBERS. 191 Gauss has shown (Disq. Arith., Art. 164), by an analysis which it is not necessary to explain here, that if f contain F both properly and improperly, an ambiguous form contained in f, and containing F, can always be assigned. This theorem comprehends the result which we have incidentally obtained in this article, that every ambiguous class contains ambiguous forms. (See also a note by Dirichlet, in Liouville, New Series, vol. ii. p. 273.) 95. The important theorem, that for every positive or negative determi- nant the number of classes is finite, is a consequence of the theory of reduc- tion. To establish its truth, it is sufficient to employ the reduction of Lagrange (Art. 92), which is applicable to forms of a positive ^ determinant having inte- gral coefficients no less than to forms of a negative determinant, and which shows that in every class of forms of determinant D there exists one form at least the coefficients of which satisfy the inequalities [2 6]s[a], [2&]^[c]. These inequalities give, if 2) be negative, a<i< —\I>, \}i\<sj--l>\ and if D be positive, \ac\<'D, \U\-^^\D. The number of forms whose coefficients satisfy these inequalities is evidently limited ; therefore, d fortiori, the number of non- equivalent classes is finite. To construct a system of representative forms of det. J), we have only to write down all the forms of det. D whose coefficients satisfy the preceding inequalities, to which we may add [«]^[t']. If the determinant be negative, it only remains to reject the forms which do not satisfy the special conditions ; if it be positive, we must examine whether any of the forms which we have written down are equivalent ; and, if so, retaining only one form out of each group of equivalent forms, we shall have the representative system required. A few particular cases of the theory merit attention from their simplicity. If i)= — 1, there is but one class of forms, represented by x'^-V^f; and by the theorems of Arts. 87 and 90, the number of representations of any uneven (or unevenly even) number by the form x** + ^/^ is the quadruple of the excess of the number of its divisors of the form \n-\-\, above the number of its divisors of the form 4n-|-3. (See Jacobi in Crelle's Journal, vol. xii. p. 169 ; Dirichlet, ihid. voL XXL p. 3. In counting the solutions of the equation x^ + y'^ = 2p, Jacobi con- siders two solutions, such as Xi' + yx = 2jp and x^ + y^ = 2p, to be identical, when a!j2 = x^, y-x = yi ; the number of solutions is thus a fourth part of the number of representations.) In particular every prime of the form 4«,-f 1 (and the double of every such prime) is capable of decomposition in one way, and one only, into two squares relatively prime ; and, conversely, every uneven number capable of such decomposition in one way only is a prime of the form 47i+ 1. 192 REPORT ON THE THEORY OF NUMBERS. [Art. 96. If D= -2, x2 + 2y« represents the only class of forms; and every uneven number can be represented by x^ + 2y\ in twice as many ways as it has divisors of either of the forms 8 n + 1, or 8 w + 3, in excess of divisors of the forms 8n + 5, or 8ft + 7. (Dirichlet, loc. cit.) In particular every prime of either of the forms 8n + l or 8ft + 3 is decomposable in one way, and in one only, into a square and the double of a square. Again, for each of the determinants - 3 and - 7, there is but one properly and one improperly primitive class, which may be represented- by the forms (1, 0, 3) and (2, 1, 2) ; (1, 0, 7) and (2, 1, 4). Uneven numbers are therefore represented by x^ + 3y^, in twice as many ways as they have divisors of the form 3ft +1, in excess of divisors of the form 3ft- 1 ; and hj x^ + 7y^ in twice as many ways as they have divisors of the forms 7ft + 1, 2, 4, in excess of divisors of the forms 7ft + 3, 5, 6. Similarly, x'^ + 4iy^ represents the only primi- tive class of det. — 4. For each of the eleven positive determinants of the first century 2, 5, 13, 17, 29, 41, 53, 61, 73, 89, 97, there is but one properly primitive class ; there is also for each of the ten uneven determinants one improperly primitive class. EepresentLng any one of these eleven numbers by D, by [T, V"\ the least solution of T^ — DU^ = 1, and by Jf, an uneven positive number prime to D, we may enunciate the theorem, ' The equation x'^ — Dy'^ = M is capable of as many solutions in positive numbers x and y, satisfying the conditions x<TVM, y<U\/M, as M has divisors of which Z) is a quadratic residue in excess of divisors of which D is a quadratic non-residue.' Thus the number of solutions of the equation x^ — ^y^^M, where if is an uneven number, and < a; ^ 3 \/M, 0<y<2 VM, is the excess of the divisors of M of the forms 8ft + 1 above its divisors of the forms 8ft + 3. The conditions <x^ TVM, <y< U VM, which are satisfied by one re- presentation, and only one, in each set, are obtained by considerations to which we shall hereafter refer (Art. 100). 96. The Pellian Equation. — The two indeterminate equations, r-DU' = l and r-DU' = 4., are, as we have seen, of primary importance in the theory of quadratic forms of a positive and not square determinant. When the complete solution of these equations is known, we can deduce, from a single representation of a number by a form, every representation of the same set ; and, from a single trans- formation of either of two equivalent forms into the other, every similar trans- Art. 96.] REPORT ON THE THEORY OF NUMBERS. 193 formation. The same equations also present themselves in the solution in integral numbers of the general equation of the second degree containing two indeterminates, and enable us in the principal case in which it admits an infinite number of solutions to deduce them all from a certain finite number. This fundamental importance of the equation T^ — DU^ = 1 was first recognised by Elder, who has left several memoirs relating to it (see Comment. Arith., vol. i. pp. 4, 316 ; vol. ii. p. 35 ; also Euler's Algebra, vol. ii. cap. vii.) ; but the equa- tion itself had already given rise to a discussion which forms a well-known passage in the scientific history of the seventeenth century. Its solution was proposed by Format (see the Commercium Epistolicum of Wallis, Ep. 8) as a challenge to the English mathematicians, and especially to Wallis. The problem was at first misunderstood by Lord Brouncker and Wallis, who each gave a method for its solution in fractional numbers ; not attending to the restriction to integral numbers implied, though not expressed, in Fermat's enunciation, without which the problem is of a very elementary character. Ultimately, however, they obtained a complete solution by a method, which WaUis describes in the Comm. Epist., Epp. 17 (postscript) and 19, and in his Algebra, capp. xcviii. and xcix., attributing it to Lord Brouncker, though he seems himself to have had some share in its invention. This method is the same as that which \a given by Eider in his Algebra, and in the first of the memoirs above cited, and which is attributed by him to Pell *. It differs, in form at least, from that now employed, and was evidently suggested by the artifices of sub- stitution employed in Diophantine problems. It is most easily explained by an example. If T^ — 13U^ = 1 be the equation proposed, the process would stand thus : — (1) 3U<T<^U; let T^3U+v,; -4:U' + QUv^+ v/ =1, (2) Vi< U <2vi; let U= Vi-f-t'a; Bvi^ -2viV2 -4V =1, (3) V2< Vi <2v2; let vi= V2-1-V3; -Sva^ -F4V2V3 +3^3^ =1, (4) V3<V2 <2Vi; let V2 = V3 -1- V4 ; . 4^3^ -2ViVi -3V =1> (5) Vi<V3<2vt; let ^3= v^+v^; - 174^ +Q,o^v^ +4^,J2 =1, (6) Qv^<Vi <7v^; let v^ = Qv^+v^; 4^52 -6V5V6 - Ve^ =1, * There does not seem to be any ground for attributing either the problem or its solution to Pell ; and it is possible that Euler may have been misled by a confused recollection of the contents of Wallis's Algebra, in which an account is given of the method employed by Pell in solving Diophantine problems. Nevertheless the equation T'^—DU'' = 1 is often called the Pellian equation after him, probably upon Euler's authority. CC 194 REPORT ON THE THEORY OF NUMBERS. [Art 96. (7) v^<Vi<2vt; let t'« = !'«+%; -Zv^^ + 2\\v^ +4v.2 =1, (8) v^<Vt< 2vs ; let v« = v,+V8; 3r,='-4r,V8 -Zv^ =1, (9) i\<v,<2v^; let Vi = v^-Vt\; -4V + 2r,r9 +3^ =1, (10) v, < Vg < 2vg ; let v^ = v^ + v^; V-6v9rio-4vio'' = l. In the last equation we may put x\ = 'i, Vio = 0; whence 7'=G49, £/'=180. It will be seen that the success of the method depends on its leading at last to an equation in which the coeflBcient of one of the indeterrainates is +1. Wallis does not prove that such an equation will always occur; and the de- monstration which he has given of the resolubility of the equation T' — DV = 1 is inconclusive. (See his Algebra, cap. xcix ; the reader will find the paralogism which vitiates his reasoning in the proof of the lemma, upon wliich it depends ; see also Lagrange's criticism in the 8th paragraph of the Additions to Euler's Algebra; and Gauss, Disq. Arith., Art. 202, note.) It is evident that the method of solution employed by Wallis really consists in the successive deter- T mination of the integral quotients in the development of j^ in a continued T . . fraction ; in addition to this, Euler observed that jj- is itself necessarily a con- vergent to the value of v/T) ; so that to obtain the numbers T and U it suffices to develope VD in a continued fraction. It is singular, however, that it never seems to have occurred to him that, to complete the theory of the problem, it was necessary to demonstrate that the equation is always resoluble, and that all its solutions are given by the development of \/D. His memoir (Comment. Arith., vol. i. p. 31 ) contains all the elements necessary to the demonstration, but here, as in some other instances, Euler is satisfied with an induction which does not amount to a rigorous proof The first admissible proof of the resolubility of the equation was given by Lagrange in the Melanges de la Soci($td de Turin, vol. iv. p. 41. He there shows that in the development of \/D, we shall obtain an infinite number of solutions of some equation of the form T' — DU^ = A, and that, by multiplying together a sufficient number of these equations, we can deduce solutions of the equation T'' — DU^ = l. But the simpler demonstration of its solubility, which is now to be found in most books on algebra, and which depends on the completion of the theory (left unfinished by Euler) of the development of a quadratic surd in a con- tinued fraction, was first given by Lagrange in the Hist, de I'Academie de Berlin for 1767 and 1768, vol. xxiii. p. 272, vol. xxiv. p. 236; and, in a simpler form, in the Additions to Euler's Algebra, Art. 37. Lastly, Gauss, Art. 96.] REPOET ON THE THEORY OF NUMBERS. 195 who in the Disq. Arith. avoids the use of continued fractions, has shown that if we form, by the method which he indicates, the period of any quadratic form of det. D, we may infer the complete solution of the equation r-DU' = l, or =4, from the automorphics of any reduced form, according as the form is properly or improperly primitive. (Disq. Arith., Arts. 198 202.) To express conveniently the principal theorems relating to these equations, we employ the following notation *. The nirmerator of the continued fraction 11 1 <7i H — 92+ 93+ -■ ?» is called the cumulant of the numbers qi, q^, ...,q„, and is represented by the symbol (q^, q^, q^, ■••,qn) '< t'he denominator is evidently the cumulant {q2, q3,...,q„). Accents are sometimes employed to indicate that the first or last quotient of a cumulant is to be omitted ; thus '(9i. g'2, 93, •••,9n) = (92, 93, •••, 9«)' ill' 92' 93. •", 9»)' = ilu 92> 93, •••' 9»-i), '(91, 92, ..., 9„)' = (92, 93, •••, 9u-i)- A penodic cumulant is represented by the notation {q^, g'2, •••)9n)j:, the suiSx indicating the number of times which the period is repeated, and a point being placed over the first and last quotients of the period. In what follows m represents 1 or 2, according as we are considering the equation r-DW = l, or =4. (i.) K Ml, /«2, •••■•i^ik be the period of integral quotients in the development of either root of a quadratic equation of determinant D, which we suppose properly or improperly primitive according as m = 1, or m = 2, the positive numbers T^ and ZZ, which satisfy the equation T'' — DU' = ')n^ are all contained in the formulae t ±f = i(^, + A,), m m B^ -2^0 «i where Tx = '(Ai, M2, • • •, ^■2k)x , -Dj;= (Mi, M2, •••,M24)a;, Ax = '(Al,A'2, ■■•,i^2kfx' * This notation is due to Euler (see Nov. Comm. Pet. vol. ix. p. 53, and the memoir already cited, 'De usu novi algorithmi in Problemate Pelliano solvendo.' Comment Arith., vol. i. p. 316). The convenient term ' cumulant ' has been introduced by Professor Sylvester (Phil. Trans., vol. cxliii. p. 474), who has also suggested the use of accents to indicate the omission of initial or final quotients. C C 2 196 REPORT ON THE THEORY OF NUMBERS. [Art. 96- and ao + 2/3o0 + ai0* = O is the quadratic equation determining the quotient ah, in which we suppose for simplicity that a, is positive. If, in particular, we consider the quadratic equation d^ — D = 0, or rather a^-D-2ad + e^ = 0, where a«<Z><(a + l)^ we have w=l, A'i = 2a, and we find, by the symmetry of the period in this case, ^x = i(-^» + ^x) = («. /*2>'*3. •••»M2l:, 2d, //2, ■ • ; i^2k)x-l, U^ = {H2, Ms. •••>M2i> 2c/, Hi, ...,M2J:)x-l> which are Euler's formulae for the solution of the equation T^ — DW = 1. (ii.) We have already observed (Art. 90) that when T^ and C/, are known, T^ and U^ are defined by the equation =D 'M- m L m J Either from this equation, or from the cumulantive formulae for T^, U^, we infer that Tj. and ZZ, satisfy the equation of finite differences, v^+2 — ^«'=e+i + Vx = 0; so that the two series, of which T^ and U^ are the general terms, are each a 2T, recurrmg series, the scale of relation being 1, -, 1. It is convenient to observe that T_y,— Ty,\ but U _.^ = — C/^. (iii) If we denote by -^ the imaginary arc y^ +Z7l^/Z) ^ m /' we have evidently — = cos>^, -. — = sm\!/-, — = = cosa;v', -. — = smx\I'. m mi m mi The analogy implied by these formulae enables us to transform many trigono- metrical identities into formulae containing T^ and ZZj. For example, from the formulae cos (<^ ± 0) = cos (f>cos6 + sin (p sin 6, sin (^ + 0) = sin ^ cos + sin cos 0, we have, putting (f> = x-^, = y^, where x and y are any positive or negative integers, i Art. 96.] REPORT ON THE THEORY OF NUMBERS. 197 (iv.) It is also found that ^ = (-i)ix(»-i) ( Ti _?Zi ?^ (_ 1)^-1 ^\ (v.) If q be any integral number whatever, we can always find a solution \Tj^, C/^] satisfying the congruences Tj^ = Tq = m, mod q, and Ux=Uq = 0, mod q. If [T;j., U^ be the least solution satisfying these congruences, X will be less than 2q, and the residues (mod 5-) of the terms of the two series T^ and U^ wUl each form a period of X terms, so that we shall always have If U^' be the first number of its series which is divisible by q, we shall have either X' = X, or 2 X' = X. In either case, the only numbers U which are divisible by q, are those whose indices are divisible by X'; and the formula T^^', — ^^ comprises all the solutions of the equation T'-^ — Dq^U^==m\ Thus, in solving the equation T'^ — DU^ = m'', we can always substitute for D its quotient when divided by its greatest square divisor. (See Lagrange, Additions to Euler's Algebra, Art. 78. Gauss, Disq. Arith., Art. 201, Obs. 3 and 4.) We may add, that if g be a prime (an uneven prime when m = 2), and if 5* and 5'' be the highest powers of q dividing U^^ and n respectively, 5* + '' will be the highest power of q dividing Z7„x. (Dirichlet, in Liouville's Journal, New Series, vol. i. p. 76.) (vi.) The methods of Lagrange and Gauss are applicable to the equation 7" — Z)Z7^ = 4, only when D = l, mod 4; because they suppose the existence of an improperly primitive form of det. D. In all other cases the equation r-DU' = 4: may be divided by 4, and reduced to the form T-DU'^1: viz. if Z) = 0, mod 4, T is even; and if Z) = 2, or =3, mod 4, T and U are both even. A similar reduction takes place if Z) = l, mod 8; the equation T^ — DU^ = 4i admitting in that case only even solutions. But if 2) = 5, mod 8, T' — DU' = 4: may or may not have uneven solutions ; and no criterion is known for distinguishing cl priori these two cases. If T^ — DU^ = 4: admit of uneven solutions, its least solution [Tj, Z/j] will be uneven; its even solutions wiU be comprised in the formula [T^n, U^nl, and consequently [^T^n, gC^s,,] will represent the solutions of T" - DU' = 1. (vii.) The equations T-DU'= -4, r-DW= -1 are not resoluble for aU values of D, but only for those values for which — 1 is capable of represen- 198 KEPORT ON THE THEORY OP NUMBERS. [Art. 96. tation by the principal form of det. Z>. Whenever the period of integral quo- tients in the development of VZ) consists of an uneven number of terms, these equations will be resoluble, and conversely. This will always happen when D is & prime number of the form 4n + l, and may happen in many other cases, but never can happen when D is divisible by any prime of the form 4n + 3. If T' — DIP= —1 be resoluble and [7\, t/j] be its least solution, the formula [T.2„+i, Uin + i] contains all its solutions, and [Tgn, C/a,,] all the solutions of T'-DU' = 1. K, in addition to the supposition that T-DW= -1 is reso- luble, we suppose that T'' — DW^ = 'i admits of uneven solutions, T^.— DU^= -4 will also admit of uneven solutions ; and if [ Ji, t/j] be its least solution, wiU represent all the solutions of T^ — DU' = —4, =4, = — 1, and =1, respect- ively. It is evident that these considerations will frequently serve to abbreviate the process of finding the least solution of T' — DIP = 1. (See a memoir of Euler's in the Comment. Arith., vol. ii. p. 35.) (viii.) The 'Canon Pellianus' of Degen (Havnise, 1817) contains a Table, giving for every not square value of D less than 1000, the least solution of the equation T^ — DU^ = 1, together with the development of %//) in a continued fraction. Its arrangement will be seen in the following specimens : — 357 97 8, (2) 4, 17 (1>1) 18, 1, 1, 33, . 180 3401 9, 1, 5, 1, 1, 1, 16, 3, 11, 8, (9, 9) 6377352, 62809633. The numbers in the third and fourth rows are the least values of U and T in the equation T^ — DW = 1. The first row of numbers is the period of integral quotients in the development of \/D : it is continued only as far as the middle quotient, or the two middle quotients, after which the same quotients recur in an inverse order. Thus, 180= (1,8,2,8,1); 3401 = (18, 1, 8, 2, 8, 1) ; 6377352 = (1, 5, 1, 1. 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1) ; 62809633 = (9, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1). Art. 96.] REPORT ON THE THEORY OF NUMBERS. 199 The numbers in the second row are the denominators of the complete quo- tients ; i.e. taken alternately positively and negatively, they are the extreme coefficients in the equations of the period. Thus the period of equations for v/357is [-33,-18,1], [1,18,-33], [-33,-15,4], [4,17,-17], [-17,-17,4], [4, 15, - 33]. The first half of the period of equations for ^/97 is [ - 16, - 9, 1], [1, + 9, - 16], [ - 16, - 7, 3], [3, 8, - 11], [ - 11, - 3, 8], [8, 5, - 9], [ - 9, - 4, 9], [9, + 5,-8], [-8,-3,11], [11,8,-3], [-3,-7,16]; the second half being com- posed of the same equations in the same order but with their signs changed. The middle coefficients of the equations are not given in the Table ; but if [ax> Px, «x + i]> [«A. + i) ^\ + u «\ + 2] be two consecutive equations, of which the former determines the integral quotient n^, they may be successively calculated by the formula Lagrange has proved that if a;^ — Dy- = H, and H he < VD, - is always a convergent to \/D ; so that a number less than -/Z) is or is not capable of repre- sentation by the principal form of det. D, according as it is or is not included among the numbers of the second row. The second Table of the ' Canon' contains the least solution of the equation T^ — DU*= —1 for those values of D less than 1000 for which that equation is resoluble. Mr. Cayley (CreUe, vol. liii. p. 369) has calculated the least solution of the equation T' — DU' — Ai, or T^ — DV = —4, for every number D of the form 8n-f-5 less than 1000, for which those equations are resoluble in uneven num- bers. This Table, as well as Degen's second Table, is implicitly contained in the first Table of the ' Canon,' as appears from the theorem of Lagrange just cited. (ix.) The theory of the equations T^ — DW = 1 and = 4 is connected in a remarkable manner with that of the division of the circle*. Let X = 2^-1-1 represent an uneven number divisible by h unequal primes, but having no square divisor ; let also the numbers less than X and prime to it be repre- sented by a or h, according as they satisfy the equation (t) = ^> ^^ \\)~ ~^ ' * See Dirichlet, ' Sur la manifere de resoudre I'dquation l^-pii!'= 1 au moyen des fonctions circulaires,' Crelle, vol. xvii. p. 286. Also Jacob! s note ou the division of the circle, Crelle, vol. XXX. p. 173. 200 REPORT ON THE THEORY OF NUMBERS. [Art. 97. and let X = be the equation of the primitive X-th roots of unity. The form of this equation (see Art. 59) implies that iaiw 26»» 2e"^ + 2e '^ =(-1)*; ' we have also the relation 2a»ir ihiv which is easily deducible from the formulae of Gauss (see Arts. 20 and 104 of this Report, or Dirichlet, Crelle, vol. xxi. pp. 141, 142). From these values iaiw 2biv 2atir 2ii'ir of Ze ^ and 2e '^ we infer that 2n(a; — e * ) and 2ll{x — e ^ ) are two quantities of the form Y+i'^^Z</\, and Y—i''^Z\/\, Fand Z denoting integral functions of x with integral coefficients; i.e. that 4X= Y^ — {—iy\Z''. From this equation, which is a generalisation of that obtained by Gauss for the case when X is a prime (Disq. Arith., Art. 357), we can deduce a solution of the 2it» equation I"-Xr' = 4. In the formula 2U{x-e~^)= Y+ii''ZV\, let us first write i for x, and then —i for i, and let us denote by X^, F,-, Zf, A'_,-, F_,-, Z_,- the values which X, Y, and Z acquire when i and — i are written for x. We thus find, denoting the number of numbers less than X and prime to it by X', 2aiir 2aiir 4^.(^-e"^)(-^•-e^ = 2^' + 2^cos2(^ + ^) =[r.. F_,+XZ, Z_,]+ VXli^'-Z, Y.^+i-^'Z., F,] ; or, writing T for i[F.. Y_i + xZi Z_,], U for i[iVZ, F.^ + i-^^Z..- F.-J and observing that Xi X_,- = 1, ^(r+ Crv/X) = 2^' n 008=^(1 + ^), r-\U' = i, where it is easily seen that T and U are integral numbers. When n is even, we may obtain a solution of the equation more simply by writing +1 or — 1 for X. (See the notes of Jacobi and Dirichlet already referred to.) It is to be observed, however, that the solution obtained by these methods is not in general the least solution. Its ordinal place in the series of solutions depends (as we shall hereafter see) on the number of classes of forms of det. D. 97. Solution of the General Indeterminate Equation of the second degree. — The solution of the indeterminate equation ax^ + 2hxy + cy'^ + 2dx-{-'2efy-\-f=0 depends on the problem of the representation of a given number by a quadratic Art. 97.] REPORT ON THE THEORY OF NUMBERS. 201 form. We confine ourselves to the case which presents the greatest complexity, that in which h^ — aG = D is a positive and not square number. The methods of solution contained in Euler's Memoirs relating to it (see Comment. Arith., vol. i. pp. 4, 297, 549, 570, vol. ii. p. 263 ; and the Algebra, vol. ii. cap. vi.) are incomplete in several respects : first, because Euler always assumes that a single solution is known, and only proposes to deduce all the solutions from it ; secondly, because it is not possible, from a given solution, to deduce any other solutions than those which belong to the same set with the given solution, whereas the equation may admit of solutions belonging to different sets ; and lastly, because he gives no method for distinguishing between the integral and fractional values contained in the formulae by which x and y are expressed. The first complete solution of the problem was given by Lagrange in his Memoir ' Sur la solution des Problemes Inddterminds du second degre' (Hist, de I'Academie de Berlin for 1767, vol. xxiii. pp. 165-311). But the following method of solution, which is different in some respects and much simpler, will be found in a subsequent memoir, ' Nouvelle methode pour r^soudre les problemes indetermines en nombres entiers ' (Hist, de lAca- d^mie de Berlin for 1767, vol. xxiv. p. 181); and in the Additions to Euler's Algebra (paragraph 7). If we multiply by aD and write p for ax + by + d, q for {h^ — ac) {y + bd — ae), Mfor (bd — aey — (b^ — ac) {d^ — of), the given equation becomes q^ — Dp^ = 31. Confining ourselves to the primitive representations of M hy q^ — Dp^ (the derived representations, corresponding to the different square divisors of M, are to be treated separately by the same method), we see that, since 2^ ^^^ ^ ^re prime, q is of the form Mr + Qp, where r and Q are two new indeterminates of which the latter may be supposed Q^ — J) <\^M\ On substituting this value for q, it will appear that N= — p^- is necessarily integral, i.e. that Q is one of the roots of the congruence ff — D = 0, mod M; and the equation will assume the form iV2)2 + 2Q^r + ifr2 = l, in which every admissible value of Q is to be employed in succession. The development of either root of the equation N +2Q6 + MB"^ = will give all the values of p and r which satisfy the equation Np'^ + 2Qpr + Mr^ = 1, because 1 is the minimum value which the form [N, O, M) can assume. (See the Additions, paragraph 2, and especially Arts. 33-35.) Or again, if we apply the transformation of Art. 92 to the form {N, Q, M), we obtain an Dd 202 REPOBT ON THE THEORY OF NUMBERS. [Art. 98. equation of the type Px^ + lQafy' + Ry^ = '^, in which Q^-PR = D, and P<^D; whence, if x"=^Px'+Qy, we finally deduce x'^-Dy^ = P, aU the solutions of which (see Art. 96, viii.) are necessarily given by the development of ^/D in a continued fraction. Applying either of these methods (the latter is not given in the Memoir, but only in the Additions to Euler's Algebra) to every equation of the form Np' + 2Q.p^ + Mr^ = l which can be deduced from the equation q^ - Dp^ = M, or from the equations of similar form obtained by replacing M by the quotient which it leaves when divided by any one of its square divisors, we obtain a finite number of formulae of the type X- § ' y S' [T, U] denoting any solution of the equation T^-DW = 1. These formulae are fractional ; but by attending to the principle of Art. 96, v., we can ascertain for each pair of formulae whether they contain any integral values or not, and if they do contain any, we can substitute for the single pair of fractional formulae a finite number of pairs not containing any fraction. The form in which the solution of this problem has been exliibited by Gauss is remarkable for its elegance. Let a, h, d b, c, e = A, d, e, f and, representing by B the greatest common divisor of 6* — ac, cd — he, ae — hd, let D ^, A ., cd — be ae — bd then, putting D'x = X+p, D'y=Y^q, we find aX' + 2bXY+cY' = D'L'. If [X„, F„] denote indefinitely any representation of Z)'A' by (a, 6, c), we have only to separate (by Lagrange's method) those values of X„, Y^ which satisfy the congruences X„+p = 0, y„ + g' = 0, modi)', from those which do not, and we shall obtain a finite number of formulae, exhibiting the complete solution required. 98. Distribution of Classes into Orders and Genera. — The classes of forms of any given positive or negative determinant D are divided by Gauss into Orders, and the classes belonging to each order into Genera. Two classes, represented by the forms (a, b, c), (a', b', c'), belong to the same order, when the greatest common divisors of a, b, c and a, 2b, c are respectively equal to those of a', b', c', and of a', 2b', c'. Thus the properly primitive Art. 98.] KEPORT ON THE THEORY OF NUMBERS. 203 classes form an order by themselves ; and the improperly primitive classes form another order. To obtain the subdivision of orders into genera, it is only necessary to consider the primitive classes ; because we can deduce the sub- division of a derived order of classes from the subdivision of the primitive order from which it is derived. The subdivision into genera of the order of properly primitive classes depends on the principles contained in the following equa- tions, in which g^ is an uneven prime dividing D, m and m' uneven numbers prime to q, and capable of representation by the same properly primitive form of determinant D. (i.) (ii.) (iii.) (iv.) (vi.) o=o- 2/ \q If Z> = 3, mod 4, (-l)|("-i) = (_i)J(m'-i), If D = 2, mods, (-l)iW-i) = (-l)i('"'^-i). If Z) = 6, mod 8, (-l)|('»-i) + i("'^-i) = (-l)|("''-i) + i('n'^-i). If D = 4, mod 8, (- l)i(-»-» = (_ i)J('»-i). If Z) = 0, mod 8, (_l)i(m-i) = (_l)i(™'-i); and (-l)^('»^-i) = (-l)i("''^-i). The interpretation of these symbolic formulae is very simple. Thus, the formula (i.) expresses that — ' The numbers prime to any prime divisor q oi D which can be represented by f, the same properly primitive form of det. D, are either all quadratic residues of J, or else all quadratic non-residues.' Again, the formula (iv.) expresses that — ' If 2) be of the form 8n + 6, the uneven numbers that can be represented by/ are either all included in one of the two forms 87i-|-l, 8n + 3, or else in one of the two forms 8n — l, 8n — 3.' All the formulae are deducible by the most elementary considerations from the three equations m = ax^ + 2 hxy 4- cy^, m = ax"^ + 2 hacfy' + cif^, {ax^ + 2bxy + C7f){ax'^ + 2hx'y' + c]f^) = { {axx' + h\xy' -f- x'y\ -V cyyy - D{xy - xy) } f Thus we find immediately /mm q / vg/ \q And again, if D = 6, mod 8, the last equation shows that axxf + h [xy + x'y] + cyy D d 2 204 REPORT OK THE THEORY OF NUMBERS. [Art. 98. is uneven ; and consequently mm' = 1-6 {xy' - ofyY, mod 8, i.e. mm' = + 3, or = + 1, mod 8, according as xy — x'y is uneven or even ; whence m and m' are either both of the forms 871 + 1, 8w + 3, or else both of the forms 8«-l, 8n-3. The form / is said to have the particular character (-) + 1, or (^)=-. according as the numbers (prime to q) which are represented by it satisfy the equation e=i- 2 or \q^ and we are to understand in the same way the expressions that / has the particular character ( — l)*</'-i)= +1, or = — 1, &c. Every particular character of a form belongs equally to all forms of the same class, and is therefore termed a particular character of the class. The complex of the particular characters of a form or class constitutes its complete or generic character ; and those classes which have the same complete chai^acter are con- sidered to belong to the same genus : so that the complete character of a form is possessed not only by every form of the same class, but by every form of any class belonging to the same genus. To enable the reader to form with facility the complete character of any given properly primitive class, we add the following Table, taken from Dirlchlet (CreUe, vol. xix. p. 338), in which /S" denotes the greatest square dividing D ; P or 2P is the quotient ^2 , according as that quotient is uneven or even ; o p, p', ... are the prime divisors of P ; and r, r the uneven primes dividing S, but not P. I. D^PS\ P = l, mod4. (a) ,S=1, mod 2. (7)' (7)'-' (/8) S=2, mod 4. ijr (7)' •'•' (7) ^=0, mod 4. (r)' (r')'-- Art. 98.] (a) S = l, mod 2. EEPOET ON THE THEORY OF NUMBERS. II. D = PS\ P = 3, mod4. (-1)'"-. (f> (#).-. (/3) ,S = 2, mod 4. (-')■"-"■ (^)' (7)--. (7) ^=0, mod 4. (-1)^'^-^^ (7)' (7)'-' (a) 5 = 1, mod 2. (_l)J(/^-u {i8) >S = 0, mod 2. III. D = 2PS\ P^l, mod4 (r)' (W'-- (w' (^)''-- (_i)i,/.-., (Z). (Z) C^)' (^). (-1)'"*-'. (^). (^) /^ ,-/■ (-1)'"-. (i). (f).- (a) <S = 1, mod2. (/3) <S = 0, mod 2. IV. D^2PS\ P = 3, mod4. '/^ // f\ (f^ \ J) / v*)'/' ' \ r /' \r' / 'P ^p f\ (f^ ■f\ (f^ (-1)1./-) (-i)i</'-. (A (^>.... (f> (f> ■p V 205 It appears from this Table, that if ft. be the number of uneven primes which divide D, the total number of generic characters that can be formed by combining the particular characters in every possible way is 2** when i) = l or 5, mod 8; 2***^ when Z> = 0, modS; and 2''*^ in every other case. But it follows from the law of quadratic reciprocity, that one-half of these complete characters are impossible ; i. e. that no quadratic form characterised by them can exist. To see this, we observe that if m be a positive and uneven number prime to D, and capable of primitive representation by f, the congruence i2'-Z' = 0, mod m, is resoluble; and consequently (—)=+!. Therefore also (^) = 1, or (— ) = 1, according as Z> is of the form PS' or 1PS\ In the first case we have 206 REPORT ON THE THEORY OF NUMBERS. [Art. 98. Q-(-l)""-'"-'. or (p(»)....(-l).<-..<i>-..; in the 6ther case ( — ) ( — ) = 1 ; i.e. (''^) = (-l)i("'-i)<i'-i) + 4('»'-»), or (^) C^) ...=(_ l)i("»-i)(P-i)+i('»''-i). A comparison of these equations with the preceding Table will show that the product of the particular characters which stand before the line of division in the Table is equal to +1 in the case of any really existing genus ; i. e. that precisely one-half of the whole number of complete generic characters are impossible. We shall hereafter see that the remaining half of the generic characters correspond to actually existing genera, and that each genus contains an equal number of classes. That genus, every particular character of which is a positive unit, is called the principal genus ; it evi- dently contains the principal class, and is therefore, in every case, an actually existing genus. Since the extreme coefficients of a form are numbers represented by it, and since, further, if the form be properly primitive, one or other of them is prime to 2 and to any prime divisor of the determinant, we see that the generic character of a form can always be ascertained by considering the values of its first and last coefficients. Thus the complete character of the form (11, 2, 15), of which the det. is-161 = -7x23 (case II. (a) in the Table), is (y)->' (25)=-'- (-i)'"---i; that of (5, 2, 33), of the same determinant, is ({)--'■ (£)--'• (-D'-'-'+i- Two forms, which have different generic characters, cannot be equivalent ; nor can a number be represented by a form if its character is incompatible with the generic character of the form. It is therefore convenient, in any problem of equivalence or representation, to begin by comparing the generic characters of the given forms with one another, or with the characters of the given numbers. The uneven numbers prime to the determinant, which are represented by fonns of the same genus, are contained in one or other of a certain number of linear forms. If R denote the product of the primes r, r', ... already defined, and if be any term of a system of residues prime to I^PR, where i is = 1, Art. 98.] REPORT ON THE THEORY OF NUMBERS. 207 when Z) = 1 or 5, mod 8, is = 3 when D = 2, 6, or 0, and is = 2 in every other case, the numbers contained in the formula I^PR + 6 can be represented only by forms belonging to that genus the character of which coincides with the character of the number 6. It is clear that one half of the linear forms, included in the formula 2''PR + d, do not satisfy the condition of possibility indicated in the Table, and are therefore incompatible with any quadratic form of determinant D ; while the remaining half of those linear forms will be equally distributed among the actually existing genera ; so that there will be either n{i(2>-l).K^-l)} or 2n{l(j9-l).i(r-l)} linear forms proper to each genus. But although no number contained in any one of the first-named linear forms can be represented by a form of determinant D, yet it is not to be inferred that every number m contained in the other half of the linear forms is capable of such representation ; for from the linear form of m, we can indeed infer the equation ( ~ ) = 1 ; but, if m be not a prime, or at least the product of a prime by a square, we cannot from this equation infer the resolubility of the congruence Q^ = D, mod m, or of any congruence of the form if = D, mod ^- We may add that if we assume the theorem that every arithmetic progression, the terms of which are prime to their common difference, contains prime numbers, the consideration of the case in which m is a prime establishes the actual existence of every genus the character of which satisfies the condition of possibility. (Crelle, vol. xviii. p. 269.) If m be an uneven number not divisible by q, a prime divisor of D, and if the double of m can be represented by an improperly primitive form J" f of det. D, we attribute to f the particular character (— )=+!, or =—1, according as ( — ) = + 1, or = — 1 ; and to form the complete character of f, we may use the Table P = l, mod4, >S = 1, mod2. p <p * All the results of this article are given in the Disq. Arith., Arts. 223-232 ; but as Gauss does not employ the symbol of reciprocity, we have preferred to follow the notation of Dirichlet. It is also to be noticed that Gauss does not use the law of quadratic reciprocity to demonstrate the impossibility 208 REPORT ON THE THEORY OP NUMBERS. [Art. 99. 99. In the preceding Articles we have briefly recapitulated the definitions and principles which constitute the elements of the theory of quadratic forms. We have hitherto followed closely the 5th section of the Disq. Arith. (Arts. 153-222 and 223-233) ; but before we proceed to an examination of the re- mainder of that section, it will be convenient to place before the reader an account of the method employed by Lejeune Dirichlet in his great memoir, 'Recherches sur diverses applications de I'analyse infinit^simale ^ la thdorie des nombres,' for the determination of the number of quadratic forms of a given positive or negative determinant. It appears from the Additamenta to Art. 306, X. of the Disq. Arith., that Gauss, at the time of the publication of that work, had already succeeded in effecting this determination ; and the method by which he effected it will at length appear in the second volume of the complete edition of his works, the publication of which is now promised by the Society of Gottingen. Never- theless the originality of Dirichlet in this celebrated investigation is unques- tionable, as there is nothing whatever in the Disq. Arith. to suggest either the form of the result, or the method by which it is obtained *. of one-half of the generic characters ; for, as we shall hereafter see, this impossihility is proved in the Disq. Arith. (Art. 261) independently of the law of reciprocity, and is then employed to establish that law. (Gauss's second demonstration, see Disq. Arith., Art. 262.) There is also an unimportant differ- ence between Dirichlet and Gauss with respect to the definition of the generic character of an improperly primitive form; for Gauss obtains the generic character (see Art. 232) by considering the numbers represented by the form, and not the halves of those numbers. But he also obsei-ves (Arts. 227, and 256, VI.) that each improperly primitive class is connected in a particular manner (to which we shall again refer) with one or with three properly primitive classes ; and that this consider- ation may be employed to divide the improperly primitive classes into genera. And it will be found that the complete character which Dirichlet's definition attributes to an improperly primitive form is, in fact, the complete character of the properly primitive class or classes with which it is connected. * The following is a list of the papers of Lejeune Dirichlet which relate to the theory of quadratic forms : — 1. Sur I'usage des s6ries infinies dans la th6orie des nombres. — Crelle, vol. xviii. p. 259. 2. Eecherches sur diverses applications de I'analyse infinitesimale k la theorie des nombres. — Crelle, vol. xix. p. 324, and xxi. pp. 1, 134. 3. Auszug aus einer der Akademie der Wissenschaften zu Berlin am 5 Marz 1840 vorgelesenen Abhandlung. (Crelle, vol. xxi. p. 98, or the Monatsberichte for 1840, p. 49.) This paper is an abstract of an unpublished memoir containing the demonstration of the theorem that every properly primitive form represents an infinite number of primes. 4. Untersuchungen fiber die Theorie der complexen Zahlen. (Crelle, vol. xxii. p. 375, or in the Monatsberichte for 1841, p. 190 ) An abstract of the following memoir. Art. 99.] REPOBT ON THE THEOKY OF NUMBEES. 209 "We propose, in what follows, to give as full an analysis as our limits will permit of the contents of the memoir. Its first section contains certain prin- ciples relative to the theory of series. (i.) 'K ^ ^ X2 ^ ^-3 < ^4 ... be a series of continually increasing positive TV quantities ; and if the ratio -r- continually tend to a finite limit a (that is to say, if, S denoting a given positive quantity, however small, we can always assign a finite value o£ n = N, such that for all values of n surpassing N the inequalities n a — S ■ k <a + S are satisfied), the limit of the expression p 2 — — , when the positive quantity p is diminished without limit, is a.' * »- « For B=QO J n = N 2 n = x P '^rTT-p = P ^TTTp + P ? N denoting a finite number ; and by virtue of the inequalities written above « = °° {a-S)^-^f »=" 1 « = " (a + ^)i + P «i +p "^ iJ. i + p "^ ,,1 + p ' n = oo J^ Observing that lim p 2 -— - is intermediate between dx una p / -— — and lim p / — -— 5. Recherches sur les formes quadratiques k coefficients et k ind6termin6s complexes. — Crelle, vol. xxiv. p. 291. 6. Sur un th^orfeme relatif aux s6ries. (Liouville, New Series, vol. i. p. 80, or Crelle, vol. liii. p. 130.) 7. Sur nne propri6t6 des formes quadratiques k determinant positif. (Monatsberichte for July 16, 1855, or Liouville, New Series, vol. i. p. 76, or Crelle, vol. liii. p. 127.) 8. Vereinfachung der Theorie der binaren quadratischen Formen von positiver Determinante. (Memoirs of the Berlin Academy for 1854, p. 99, or, with additions by the author, in Liouville, New Series, vol. ii. p. 353.) 9. Demonstration nouvelle d'une proposition relative k la theorie des formes quadratiques. — Liouville, New Series, vol. ii. p. 273. 10. De formarum binarium secundi gradus compositione. — Crelle, vol. xlvii. p. 155. The three last papers contain important simplifications of theories which appear in a very compli- cated form in the Disq. Arith. To two of them we have already referred (Arts. 93, 94). * This theorem is a generalisation of that in the memoir (Crelle, vol. xix. p. 326). It is given by Dirichlet in No. 6. of the preceding list. £6 210 REPORT ON THE THEORY OF NUMBERS. [Art. 99. and is consequently unity, we infer from the last inequalities that and therefore also lim p 2 , ,^ . n-l "^n which is identical with it, because lim p 2 —rTn — ^ differs from o by a quantity comminuent with S ; i.e. n— 00 J n = l "-n since by hypothesis 5 is a quantity as small as we please. (ii.) A convergent infinite series may be convergent in two very different ways. It may be convergent, and always have the same sum irrespective of the arrangement of its terms ; or it may be convergent for certain arrangements of its terms, giving the same or different sums for these different arrangements, and divergent for other arrangements. We suppose, however, that we con- sider only such different arrangements of the terms of a series as are compa- tible with the condition that any term which occupies a Jinitesimal place in any one arrangement should occupy a Jinitesimal place in every other arrange- ment *. Thus the series 1 1 1 li + P + 21 + P "^ 31 + P "*"■■■' ^^ ' is convergent, and has the same sum in whatever order we sum its terms ; but of the two series * This condition is necessary, because without it the sum of no series whatever would be inde- pendent of the arrangement of its terms, if by the sum of a series we understand the limit to which we approximate by the continual addition of its terms in the order in which they are given. For example, the series cited in the text, 1 1 1 + 7n+i + ^i+i + -' P>^' is convergent, and its sum is irrespective of the arrangement of its terms, provided that arrangement satisfy the condition enunciated in the text. But if we were to arrange the terms of the series in an . order regulated (say) by the niunber of primes dividing their denominaton., the limit to which we should continually approach by adding together the terms taken in their new order would be S ,^ » 1 P' in which p denotes any prime, and not 2 -^^^ > in which n denotes any integer, Art. 100.] REPORT ON THE THEORY OF NUMBERS. 211 1 1 — + -T 2^ 3^ 1 1 — + — 4? 5? -Ta +•••. . 1111 1 + — r + — + — 2^ 5^ 7^ JL 1 - ^ + 3? <2p. 5^ 1^ 41 only the first is convergent ; while the two series , 11111 2^3 4^5 6^ ' , 11111 ^ + 3-2 + 5 + 7-6 + - axe both convergent, but have two very different sums *. These observations will show the importance of the following proposition f : — ' If c„ be a periodic function of n, satisfying the equations Ci + C2 + C3 + . . . + Ci = 0, »=co g the series 2 — in which the terms are taken in their natural order, is con- vergent for all values of s superior to zero, and its sum is a continuous func- tion of s.' For if we add together the h consecutive terms , + 7r- .+ •••+: {km-Viy ' {km + 2)' ' '" ' (km + k)'' we obtain a fraction of which the denominator is of the order ks in respect of m, whUe the numerator is only of the order (^— 1) s — 1, because the coefficient of m'*~i'' is zero. We may therefore replace the given series by a series of , in which (p{m.) is a function of the order l+s in respect m = =o the form 2 =1 ^(to) of m. This series is always convergent for positive values of s ; its convergence is irrespective of the arrangement of its terms, and its sum is a continuous fiinction of s, because </)(m) is a continuous function of s. The given series is therefore also convergent, and its sum is a continuous function of s. 100. The second section of the memoir refers to the symbols of reciprocity of Jacobi and Legendre (Arts. 15, 16, and 17 of this Report). * These illustrations are taken from the Memoir on the Arithmetical Progression in the Berlin Memoirs for 1837, pp. 48 and 49. t The demonstration in the text is a little simpler than that given by Dirichlet, who uses the fonction F to express the sum of the series. £62 212 REPORT ON THE THEORY OF NUMBERS. [Art. 100. The third and fovirth sections contain the principal theorems relating to the generic characters of quadratic forms, and to the representation of numbers. There is only one of these theorems to which we need direct our attention here, as the others have already come before us in the preceding articles. Let (a, 6, c) be a primitive form of the positive determinant D ; let also (a, h, c) {Xo, yi)' = M a positive niunber represented by (a, h, c); m the greatest common divisor of a, 26, c; [T, U"\ the least positive solution of r-Dir = m'; so that if the two formulae [a!;„, 3/,,] and [ — £p„, — 2/,,] will together express every repre- sentation of M, which belongs to the same set as \xq, y^. Similarly, let \x\, «/'„], [ — «'„, -2/'»] denote a complete set of representations of the positive number M' by (a, h, c). K we trace the hyperbola represented by the equation ax^ + 2hxy-\-cy'^ = l referred to rectangular axes, the diameters included in the formula y——x, in Xjg which ^ is to receive all values from — 00 to +00, will form a pencil of lines, which all meet the curve, and which, commencing with the asymptote y= — -j: — r X, continually recede from it, and approximate to the asymptote y = -jj: — , X. The sectorial area contained between any two consecutive lines of this pencil and either branch of the hyperbola is constant and equal to ^ . —Tj; log ^^ ; as may be ascertained by employing polar coordinates. Since the same observations apply to the pencil y = ^ a?, we infer that the lines of these two pencils lie alternately, unless the two pencils coincide. Let us now suppose that in the form (a, h, c), a is positive and c negative ; so that the axis of x does, and the axis of y does not cut the curve. On this supposition the values of ^^ and of ^ continually increase from -=- — j- x„ flj „ vD + o to ,r,_i 3S n increases from — 00 to +00. The alternate position of the lines of the two pencils gives, in this case, the theorem, — 'The inequalities y\^yn^y\.r X ]g X^ ^ t + 1 Art. 100.] EEPORT ON THE THEORY OF NUMBERS. 213 in which h represents any given number, are satisfied for one value of n, and one only.' K, taking a for M' and [1, 0] for [x'o, y'o], we put ^ = 0, we obtain the conclusion, — ' Each set of representations of the positive number M by the form (a, h, c), in which a is positive and c negative, contains one and only one representation which satisfies the inequalities a;„>0, yn>0, yn^ j'-hU '^"' It is in this form that the theorem appears in Dirichlet's memoir. We may add that any values of x and y which satisfy these inequalities will give a positive value to (a, b, c) ; for such a pair of values will coiTespond to a point situated in the internal angle between the asymptotes of the hyperbola. The fifth section contains the demonstration of the theorem, that if A denote the absolute value of B, and yff (2 A) be the number of numbers less than 2 A and prime to it, a properly primitive form of determinant D will acquire a value prime to 2D, if its indeterminates x and y satisfy any one of a certain set of 2A\|^(2A) congruential conditions included among the 4A2 conditions represented by the formulae a; = a, mod2A; y = (3, mod. 2 A, in which both a and /3 represent any term of a complete system of residues, mod 2A; but will acquire a value not prime to 22), if a; and y satisfy any of the other congruential conditions. If the form be improperly pi'imitive, the number of congruential conditions that will render its value unevenly even and prime to A will be A -vl/- (A), or 3 A vf' (A), according as i) = 1, or =5, mod 8. These theorems are easUy demonstrated by considering separately the prime divisors of A. For example, if the form (o, b, c) be improperly primitive, and 2> be a prime divisor of D, since either a or c is prime to p, let a be prime to p ; then (ax + byf — Dy^ will be prime to p, when ax + by is so ; i.e. it will be prime to p, for p{p—V) combinations of the residues (mod p) of X and y ; or, if ^9" be the highest power of ^^ dividing D, for ^j^n-i (^j-l) combinations of the residues of x and y, mod p". Again, the 4 combinations of residues for the modulus 2 will give ^ {a, b, c) the values 0, ^a^, Jc2, ^a + b + ^c, of which it is easUy seen that one or three will be uneven, according as ac = 0, 214 REPORT ON THE THEORY OF NUMBERS. [Art. 101. or 4, mod 8; i.e. according as D = l, or 5, mod 8. The combination of these results will give Dirichlet's theorem. 101. Series expressing the number of Primitive Classes. — The sixth section of the memoir contains the demonstration of the formulae which express in the form of an infinite series the number of classes of properly and improperly primitive quadratic forms of a given determinant. We shall abbreviate the demonstration of these formulae by using the theorem of Art. 87. Let h be the number of properly primitive classes of determinant D ; we shall first suppose D to be negative, and = — A ; let also be a system of forms representing the properly primitive classes of that deter- minant ; and let us consider the sum S = {diX^ + 2 bixy + Ciy^y ^ (a^x^ + 2 h^xy + c^y^)' '" " (aj,x^ + 2hj,xy + c^y'')' ' the sign of summation 2j. extending to all values of x and y from — oo to + 00, which give the form (%., \, Cj.), a value prime to A. By the theorem of Art. 87, any uneven number n prime to A is capable of 21. (-j\ repre- sentations by the properly primitive forms of determinant D (for there are 2 (-t) sets of representations, and each set contains two *.) We have there- fore the equation ^=2 2[2(^)^] (a) (the inner sign of summation referring to every divisor d o? n; and the outer sign extending to every positive value of n prime to 2 A). If we write n for d, and nn for n, so that n and n each represent any positive number prime to 2 A, this equation assumes the simpler form ^=22(:^)-\, (6) V w / (wn )' ^ ' the sign 2 indicating two independent summations with respect to n and n' ; or, if we perform the two summations separately, and omit the accent, ^=22-l2(:^)^ (c) * If A = 1, each set contains four representations. To obtain a correct result in this case, we must therefore double the right-hand members of the equations (o), (6), (c), and (A). Art. 101 ] REPORT ON THE THEORY OP NUMBERS. 215 To deduce an expression for h from this equation, we write 1 + /> for s, and multiplying, each side by p, we suppose p to be positive and to diminish with- out limit. In order to find the hmit o£ pS on this supposition, we consider separately the partial sums, such as P^ (^^2 + 26a:v + CT''V + p ' ^^ ^^^^^ ^^ '^ composed. ^ !f if J If j-r— be the n-th term of the series 2 - — —-— , ■A,4.n , in which we suppose that the terms are so arranged that no term surpasses any that precedes r ^ Wta • For if 2 A^ + ^„, 2 A,? + ^„ represent it, it can be shown that lim generally any one of the 2A\|/-(2A) systems of values that can be attributed to X and y consistently with the condition that (a, b, c) assumes a value prime to 2 A, the number of terms up to k„ inclusive (i.e. the number n) is evidently equal to the number of points having coordinates of any one of the forms [2A^+^(,, 2Ai; + i7j that lie within the ellipse ax^ + 2bxy + cy'^ = k„, together with one, or aU, or some of the similar points lying on the contour of the ellipse, according as , ^^ is the first or the last, or neither the first nor the last of . . . . . vk the terms equal to it in the series. The area of the ellipse is —~ ; whence, if n be very great, the number of the points we have defined is approximately 2Ax|.(2A)x)t„ iAV-^ , the error being of the same order as v'^,, ; i.e. lim n ■^{2 A) k^ 2A^/A Hence by Dirichlet's first Lemma (Art. 99), Again, by the same Lemma, the expression p2 n^ + p has — „ . for its limit. 2A when p diminishes without limit. And, lastly, the limit of the series S C — j —^ is the series 2 ( — ) - , in which the terms are taken in their natural order. To establish this, we observe that the symbol ( — ) is a periodic function of n, and that the sum of the terms of which one of its periods is composed is zero. Using the notation of Art. 98, and attributing the value +1 or — 1 to the symbol «^ according as P = l or =3, mod 4, and to the symbol e according as 216 REPORT ON THE THEORY OF NUMBERS. [ah. 101. D=sP5* or = 2PS'', we have, by Jacobi's law of reciprocity, Hence (— )= (~t)> '^ n = n', mod 2^ PR*; or ( — ) is a periodic function of n. Again, if a and b denote the general terms of a system of residues prime to 2* and p respectively, we find 25J<»-i'€i(»'-i' (^) = 2^l(''-i)€*(<''-i) X n . 2 (-) X n (r- 1), Jr where in the left-hand member the summation extends to every value of n prime to ^J'PQ and less than it, while in the right-hand member the signs of sum- mation refer to a and h, and the signs of multiplication to p and r respectively. This equation is easily verified ; for if n = a, mod 2^, = h, mod p, = h', mod p', ..., we have ^n^ ,1,^ ,h\ ^|(n-:),i(n-l) (^) = ^l(a-l),i«.-:) (A) (^ ... ; so that each member of the equation consists of the same units. But one at least of the factors of which the right-hand member is composed is zero ; unless we have simultaneously ^ = 1, 6 = 1, -P = l, a supposition which is inad- missible, because it implies that Z) is a perfect square. We infer therefore that 2 (— ) = 0, i. e. that the sum of the terms of a period of the symbol (— j is equal to zero. If, then, we suppose the terms of the series 2 ( — ) — — — to be taken in their natural order, it wiU follow from Dirichlet's second Lemma (Art. 99) that its sum represents a finite and continuous function of p for all values of p superior to — 1 ; i. e. the limit of the series 2 (—^ , for (0 = is the series 2 (--) - , in which the terms are taken in their natural order. We thus obtain the equation "n/ n fiV- (A) Vn/ n * The index h is not the same as in Art. 98 ; it is 1 when 8 = 1, « = 1 ; 2 when 6 = — 1, e = 1, and 3 when e = — 1. Art. 101.] REPORT ON THE THEORY OP NUMBERS. 217 Secondly, let the determinant D be positive ; and let us retain the same notation as in the former case. If in the series S= (in which it is convenient to suppose that the forms (%, b^, cj.), representing the properly primitive classes of determinant D, have their first coefficients positive, and their last coefficients negative) we suppose the sign of double summation 2^ to extend only to those integral values of x and y which render the value of the form (%, &j, Cj) prime to 2D, and which further satisfy the inequalities x> 0, y > 0, y S T-hU x. we obtain, by a comparison of Arts. 87 and 100, the equation S=AH-)->-> (c) n' \n^ n' ^ ^ in which n denotes any positive number prime to 22), and which corresponds to equation (c). 1 If k'*!" be the n-th term of the series {ax^ + 2bxy + cy^y*f'' n is equal to the number of points having coordinates of any one of the forms [2A^ + eo, 2Ar, + „,l which lie in the interior of the sectorial area, bounded by the positive axis of x, the arc of the hyperbola ax^ ■\-2bxy + Gy'^ = k„, and the straight line aU y'T-bU"^' together with one, all, or some of the similar points on the contour of the sector. The area of the sector is h ^log(r+£^^i)); whence, reasoning as before, we find 2VD h = (-)- (B) log [T+ Us/D] \n/ n for the number of properly primitive forms of a positive determinant B. The corresponding formulae for improperly primitive forms are obtained by a pre- cisely equivalent process. The results are, if Z) = — A, 218 REPORT ON THE THEORY OF NUMBERS, [Art. 102. [2-(-l)i(z>-)]A'= |^A2(^)1*. (C) and if D= +A, P-(-i)"''-"]y° iog^(r-fpvi>) ^(¥)«- •■ • • <!') [T', If] denoting the least solution of the equation T - DIP = ^. 102. Proof that each Genus contains the same number of Classes. — The sixth section of the memoir also contains a demonstration of the proposition to which we have already referred (Art. 98), that all the possible genera actually exist, and contain an equal number of classes. This demonstration is not deduced from the expression for the number of properly primitive forms, but depends on an equation between two intinite series similar to the equation (a) of the last article. Let x denote any one of the particular characters proper to the determinant, and let (f> be any term in the product IT (1 + y), with the exception of the first term, which is vinity, and also of that particular com- bination of the values of x> the value of which, by the condition of possibility, is also a positive unit. If \ be the number of particidar characters, 2^ — 2 will be the number of expressions symbolised by (f>. Let H and H' be the numbers of classes satisfying the conditions <^ = 1 and ^ = — 1 respectively. It can be shown, as follows, that H=H'. Confining ourselves, for perspicuity, to the case of forms of a negative determinant, we have, by the principle of Art. 87, V ^1 .2 ^ + +2 '^ft ^(a^x^^-U^xy + c^yy ^ (a^x^ + 2b.,xy + c.,y''y '" '' {a^x^ + 2bkxy + c^yj where in the right-hand member (— ) is -f-1 or —1, according as the number n satisfies the condition ^ = 1 or^= — 1; and similarly, in the left-hand member ^= — 1 or = -1-1, according as the generic character of the form (aj, 6j., c^) satisfies the condition <^ = 1 or^=— 1. In this equation the signs of sum- mation have the same signification as in the similar equation (a) of the last article ; and, as in that equation, the right-hand member may be expressed in the simpler form \n/ n* \ny \n/ n' * If A =3, we must triple the right-hand member of this equation ; as each set of representations of a niunber by a form of determinant — 3 contains six representations, instead of two. Art. 103.] BEPORT ON THE THEORY OP NUMBERS. 219 K we now write 1+ p for s, and, multiplying by p, allow p to converge to zero, the limit of the left-hand number is (H—H') ^.^ ,^ • The series 2 ( — j T— j "Y^ converges to a finite limit ; for ( — J and f^— j are each of -j^Y S and e denoting positive or negative units, and Q an uneven number composed of unequal primes dividing D ; their product is therefore another expression of the same form, in which S, e, and Q are not simultaneously equal to 1, because we have expressly excluded that combination of the particular characters which causes (— ) to coincide with ( j • It can therefore be shown, by reasoning as in the last article, that the second Lemma of Art. 99 is applicable to the series, and that it converges to the finite limit 2 ( — j (~j -• Similarly, it may be shown that 2 (—j — ^p converges to a finite limit. The limit of the right- hand member of the equation (d) is consequently zero on account of the evanescent factor p; from which it follows that H=H'. Let G^, G^,... be the difierent possible genera; h^, h^,... the number of classes they severally contain ; (yr) the value of <p for the genus G. The equation H— H' = comprises 2*^ — 2 equations of the type (|_)*,.(|)A.+ ...=0. corresponding to the 2^-2 different expressions symbolised by (p. If we multiply each of these equations by the coefficient of Aj in it, and add the products to the equation 2Ai-(- 2^2-1- 2h^+ ... = 2h, we arrive at the conclusion 2^Aj. = 2A. For the coefiicient of h^ in the result- ing equation is the product 11 1 -|- C^) C^) ! and this product is 2*', if G^ and G^ are identical, but is zero in every other case, as one at least of the factors will be zero. 103. The seventh section (Orelle, vol. xxi. p. 1) commences with the proof of the theorem that the number of sets of representations of any number M F f 2 220 REPORT ON THE THEORY OF NUMBERS. [Art. 103. prime to 22) by quadratic forms of determinant D, is equal to the excess of the number of those divisors d oi M which satisfy the equation 3Jw-i)e*«P-i)(-^) = l, above the number of those divisors which satisfy the equation the symbols S and e having the same signification as in Art. 101. Of this theorem, which coincides with that of Art. 87, since two demonstrations are given, one purely arithmetical, the other derived from ■ the equation (6) of Art. 101, the proof of which in Dirichlet's memoir does not involve the theorem of Art. 87, but is deduced from the arithmetical principles on which that theorem itself depends. We have already referred (Art. 95) to some of the particular results which can be deduced from the general theorem. It is evident from the mode of formation of the equation (6), or of the corresponding equation for a positive determinant, that it may be generalised by taking instead of the power {ax^ + 2hxy + cy)~', any function of ax^ + 2 hxy + cy^ which renders the two members of the equation convergent ; i. e. we may write, in the case of a negative determinant, 2i . (^ (oi a;'' + 2 6i a;?/ + Ci 2/**) + 2^ . <^ (a^ x'' + 2 6, xy + C;, 3/=) + . . . = 2 2 (— ) (^ (wft). Dirichlet illustrates this observation by giving to </> the exponential form (f, which satisfies the condition of convergence, if the analytical modulus of q be inferior to unity. Each double sum, such as 'Z,^'' + ^^^-^'^- in the left- hand member of the equation can then be replaced by 2aA\|/-(2A) (or sometimes by fewer) products of the "" *-" i(2aA» + »o)» » = " ^(2aAr + r,)'' ^ q" X 2 5« in which each simple series such as r "=" ^(20Ai, + t)„)» 2 a" 1)=_00 Art. 103.] REPOET ON THE THEORY OP NUMBERS. 2*21 can be expressed by meaxis of the elliptic function 9 ; the right-hand member can also be expressed by means of elliptic series. K, for example, Z) = — 3, we have the equation V=O0 V= + 00 V= — OO V= —00 »= + OO » = 00 v3(2»+l)2 fc=oo = 2 k = v6fc+l + q' 5(6fc + l) 1c = co q6J; + 5 I ^S(6fc + 6) - 2 Jc = It does not appear that this remarkable transformation, which is only very briefly noticed by Dirichlet, has been further examined. (See a note by Mr. Cayley in the Cambridge and Dublin Mathematical Journal, vol. ix. p. 163.) In the eighth section Dirichlet assigns the relation between the numbers of properly and improperly primitive classes. When the determinant is ne- gative we find, by a comparison of the formulae (A) and (C), h = h', or h = 3h', according as D = l, or =5, mod 8 ; observing only that if Z)= — 3 we have, exceptionally, h = h'. When the determinant is positive, we infer from the formulae (B) and (D), log UT'+U'Vl)) y log{T+U^/D) ' or 3logi(r+t7VZ)) log(T+UVD) ■ according as Z) = l or =5, mod 8. Comparing these expressions with the observations in Art. 96 (vi.), we find, if Z) = 1, mod 8, h = h' ; and if Z) = 5, mod 8, h = h', or h = 3h', according as the least solution of the equation J" — Z)t/^=4 is uneven or even. Dirichlet also deduces from the formulae (A) and (B) the relation which subsists between the numbers of properly primitive classes for any two deter- minants which are to one another as two square numbers. It is sufiicient to consider two determinants such as D and BS', of which the former is not divisible by any square. If h and H be the numbers of classes for these two determinants, we have evidently, when the determinants are negative, n^ n (-)-■ 222 REPORT ON THE THEORY OF NUMBERS. [ah. 104. the two series in the numerator and denominator not being identical, because in the one n is any number prime to 2DS'', in the other any number prime to 2D. But, by a principle due to Euler, p representing any prime, except those dividing 2DS'^ or 2D. Hence /r.*sn(i-(|)l). if s denote any prime dividing (S but not dividing D.' For a positive deter- minantwefind D. log (T+U^D) ^ = ''^^^\^-(V)log(T'+UWDy [T'f V] denoting the least solution of the equation T* — DS^U* = 1; i.e. the least solution [T^, C/j] of T^ — DU^ = 1, which satisfies the condition U^=0, mod S ; so that we may write In a subsequent note (No. 7 in the list) Dirichlet infers from this expres- sion that, given any positive determinant D, we can always deduce from it an infinite number of determinants of the form DS^ having each the same number of classes. For if we attribute to S a, series of values of the form n . s", all composed of the same prime numbers s, and having continually increasing numbers for the indices of those primes, it appears from a remark S to which we have already referred (see Art. 96, (v.)), that the quotient y will eventually be constant ; i. e. there will exist an infinite series of determinants, all composed of the same primes, and all having the same number of properly primitive classes. As it is possible to find determinants contained in a series of this kind, and having only one class in each genus, it appears that the number of the positive determinants, which have only one class in each genus, is infinite. This result, which was anticipated by Gauss (Disq. Arith., Art. 304), is remarkable, because it is probable, from the result of a very extensive induction, that there are but 65 negative determinants, of which the greatest is — 1848, having the same property. 104. Summation of the series expressing the number of Properly Primitive Classes. — It appears from the last article that, to obtain expressions in a finite Art. 104.] BEPOET ON THE THEORY OE NUMBERS, 223 form for the number of classes, we may confine our attention to the order of properly primitive forms, and may suppose that the determinant is not divisible by any square. To sum the series 2 C — ) - upon this supposition, Dirichlet employs the formulae given by Gauss in his memoir, ' Summatio Seriemm quarundam singularium,' to which we have already referred in this Report (Art. 20). The ninth section is occupied with the demonstration of these formulae ; in the tenth they are applied to the summation of the series 2 f — j - • Two different methods are given by Dirichlet, by either of which this summation can be effected. (i.) If A; be the index of periodicity of ( — Y so that (A) = (-). ^^d 2(^) = 0, h the summation indicated by the symbol 2 extending to all values of n prime 1 to 22) from 1 to h, we have, writing V for 2 ( — ) - , ./o 35* - 1 where f(x) = 2 ( — ) a;", so that /(I) = 0. Integrating by the ordinary method of decomposition into partial fractions, we find -yfcF= 2 1 / ^"^'^' \ /.I dx 2mirt x — e To simplify this complicated expression, it is requisite to transform the symbol ( — J by the law of reciprocity, and to consider separately the eight cases which arise from every possible combination of the hypotheses, (a) D positive or negative, {^) D even or uneven, (7) D, or \D, =1, mod 4, or =3, mod 4. As an example of the process, we shall take the two cases in which D = 3, mod 4, so that 224 REPORT ON THE THEORY OF NUMBEB8. [Art. 104. 2mwi A still denoting the absolute value of D. The value of /(e *^ ) is assigned by the formulae of Gauss ; it is or zero, according as m is, or is not, prime to 4 A *. We thus find * If j> be any prime divisor of A, an uneven number admitting of no square divisor, and if, for brevity, P = — ; we have, by Gauss's formula, according as m is or is not prime to ]>. If we multiply together the equations of this type, cor- responding to every prime divisor of A, and observe (1) that = 2 . A/"* represents a system of residues prime to A, (3) that n (— )ii(p-i)' = (- l)21(Pi-i)(/'2-i) X tSKP-i)" = il[S(p-i)P = ,-l(A-i>»^ a iSmtri we find ^(x)* ^ =(^)i«'^-i)VA, or = (2) according as m is or is not prime to A. We have already met with this equation in Art. 96, (ix.). If in the equations (1) we write 4P for P, and join to them the equation i;(_l)J(i-i)ei*'»A^i_2i(_i)i('»-J)+4(A-i), («i uneven), or =0 (w even), in which k is either term of a system of residues prime to 4, we obtain after multiplication the equation which is employed in the text. And similarly may the function f(e * j be evaluated. whatever be the form of D. The formulae (A) and (A') of Art. 20 are only particular cases of the general result obtained by Gauss in the ' Summatio Serierum &c.' The general formula, including (A), is k=n-l ,h. = ^n'' h h denoting any number prime to n. When n is even, the formula (A') of Art. 20 is similarly include d in the following, „ Sr^>"=0, or =(|^)t-l(*-'0"(l+i)V^ {=(|-)(l + »*) V^}' according as n is unevenly or evenly even. Wlien n is uneven and not divisible by any square, the two sums 2 r**' and 2f— )r*» * = o ^M'' are identical, as appears from a comparison of (2) with the generalisation of (A), and has been already observed in the case when n is a prime (Art. 21). {[Aug. 8, 1877.] The generalised formulae (A) and (A') here given coincide with the formulae Art. 104.] REPORT ON THE THEORY OF NUMBERS. 225 the suromation extending to all values of m prime to 4 A and less than it. In -j~j is zero, because the terms correspond to m and 2 A + m destroy one another ; so that -4AF=2^•iU..)VA2(-l)l(.»-)Q[log(sinfJ)-^•fJ]. Distinguishing the two cases D = A, and D = — A, and observing that the imaginary parts vanish identically, as they ought to do, because V is real, we have, finally, if D = A, -4DF=2,/fl2(-l)!<-"Qlogsm(^) andtfZ)=-A, _ ,^. -4AF = ]m 2s/A Vm/ 2s/A m. 'D^ (ii.) The series 2 { — j - can also be summed by substituting for ( — j its trigonometrical value deducible from the formulae of Gauss. We will take as an example the case in which D= — A = 3, mod 4. Writing n for m, and m for n, in the equation J 2 m IT In. /(e^^; = 2ii(i + A)'(-l)i("'-i>(^)v/A, we find, observing that ^(1 + A) is uneven, 1 ^/D\ . /2mmr\ = 27A^(m)'^n^A-)' imniri 4A of M. Lebesgue (Liouville (I), vol. xii. p. 509). If w = P = aj>, h = Q = bq, a and 6 being powers of 2, p and g uneven, we have this is wrong when P is even, and p = — 1, mod 4. "We have, however, in every case. 226 REPORT ON THE THEORY OF NT7MBERS. [Art. 104. the summation extending to every value of m prime to 4 A and less than it. Substituting this expression for / — ) in V, we have _ 1 „ /D\ „ 1 . /2inmr\ 2 VA Vm/ n V 4A / Since the expression which we have substituted for / — ) is zero, when n is not prime to 4 A, we may attribute to «, in the series _ 1 . /2mmr\ 2 - sm ( . ■ ) , n V 4A / either all uneven values, or all int^ral values. The sum of the series since sinSaj sin 5a; is, by a known theorem, :j7r or —^tt, according as 0<a;<x, or ttkxk'Ztt. Hence attributing to n only uneven values, and denoting by m' and w" the values of m inferior and superior to 2 A, V= TT X>N /D 8VA 4 /A \m because (|') = " (2AT^')- If we attribute to n all integral values, the equation ,, . sin 33 sin 20! sin 3a: i(x-a;) = -J— + — 2— + — 3— + ..., which subsists for all positive values of x less than 2rir, will give the value already obtained for V by the former method, viz., _4AF=-^2(^)m. 2 -v/A Vm/ The mode of application of this method may be still further varied ; for, —J, we may leave the factor (-l)^*"-!' j-j, by means of the equation /n\ 1 ^ /m\ /2mnv\ ( a) = 7a ^ (a) ^'^^ (-A-)' Art. 104.] REPORT ON THE THEORY OF NUMBERS. 227 which, as well as the substitution which we have employed, is deducible from the formulae of Gauss *. We should thus obtain a third expression for V, different in form from both of those which we have already found. The forms which the expression of h can assume are very numerous ; we select the following as examples, Z> still denoting a determinant not divisible by any square. I. If Z) = l, mod 4. For a positive determinant, D = A, ^D /imr\ h = (4) ,~ 2 (i^) log tan (^ For a negative determinant, J9 = — A, the summations extending to every uneven value of m prime to A and less than A. II. K D be not = 1, mod 4. For a positive determinant, 1 h=- \7n) °^ . /TO7r\ log {T+UVD) For a negative determinant, the summations with respect to m and m extending to all values prime to 2 A, and inferior to 4 A and 2 A respectively. Dirichlet observes that when the determinant is positive, the coefficient of logTr+TTTD) ^ ^ logarithm of the form log {T^+UuVD); (T^, C/,,) being one of those solutions of the equation T^ — DU' = 1 which are deducible from the theory of the division of the circle. Thus h is in fact determined as the index of the place occupied in the series of solutions of I" — DU' = 1, by an assigned trigonometrical solution. (See a note by M. Amdt in Crelle, vol. Ivi. p. 100.) * See equation (2) of the preceding note. Gg2 228 REPORT ON THE THEORY OF NUMBERS. [Art. 104. In the particular case in which the determinant is a prime of the form 4n + 3 taken negatively, an expression for the number of classes had already been given by Jacobi (Crelle, vol. ix. p. 189). It would seem, from his note on the division of the circle (Crelle, vol. xxx. p. 166), that the unpublished method, by which his result was obtained, formed a part of that theory. VIII. REPORT OK THE THEORY OF NUMBERS. Paet IV. [Report of the British Association for 1862, pp. 503-526.] 105. (jrENERAL Theorems relating to Composition. — The theory of the composition of quadratic forms occupies an important place in the second part of the 5th section of the ' Disquisitiones Arithmeticse,' and is the foundation of nearly all the investigations which follow it in that section. In accordance with the plan which we have followed in this portion of our Report, we shall now briefly resume the theory as it appears in the * Disquisitiones Arithmeticae,' directing our special attention to the additions which it has received from subsequent mathematicians. We premise a few general remarks on the Problem of composition. If Fi{xi, x.^, ...,x^ be a form of order m, containing n indeterminates, which, by a bipartite linear transformation of the type a = l, 2, 3, ...,«, /8 = 1, 2, 3. ...,7i, 7 = 1, 2, 3, .,.,», is changed into the product of two forms Fi{yi, 3/2> •••>2/n) ^^^ -^sC^^d ^, •••>z„) of the same order, and containing the same number of indeterminates, F^ is said to be transformable into the product of F^ and F^ ; and, in particular, if the determinants of the matrix a. ■a.fi,y which is of the type « x «^ be relatively prime, F^ is said to be co7npounded of Fi and F^. Adopting this definition, we may enunciate the theorem — ' If F^ and dx^ dZy a = l,2,3,...,n,) ! = 1, 2, 3,...,«. ) : = 1, 2, 3, ...,71,1 ' = 1, 2, 3,...,m.j 230 REPORT ON THE THEORY OF NUMBERS. [Art. 105. be transformable into F^xFa, and if Fi, G^, G^ be contained in Gi, F^, F^ respectively, (r, is transformable into G^Y-G^] and, in particular, if F^ be compounded of F^ and F^, and the forms F^, G^, G^ be equivalent to the forms (tj, F^, Fs respectively, G^ is compounded of G^ and G^.' It is only in certain cases that the multiplication of two forms gives rise to a third form, transformable into their product. Supposing that F2 and Fs are irreducible forms, i.e. that neither of them is resoluble into rational factors, let /i, I2, Is, be any corresponding invariants of F^, F^, Fs, and let us represent by B and C the determinants dyp /3 = « = 1, 2, 3, 7- The transformation of F^ into F2 x Fs then gives rise to the relations mi I^xB^^I^x Fs\ mi /ixC" =/3xi?;', i denoting the order of the invariants /j, I^, Is- If one of the two numbers /a and /g be different from zero, we infer that m is a divisor of n. For if - be the fraction — reduced to its lowest terms, the equations /i" X 5'" = // X i^a"', I^" X Ci** = Is" X F^"* imply that F^ and Fs (cleared of the greatest numerical divisors of all their terms) are perfect powers of the order yu ; i.e., m = 1, or m divides n, since F^ and Fs are by hypothesis irreducible. We thus obtain the theorem (which however applies only to irreducible forms having at least one invariant dif- ferent from zero) — ' No form can be transformed into the product of two forms of the same sort, unless the number of its indeterminates is a multiple of its order.' For example, there is no theory of composition for any binary forms, except quadratic forms, nor for any quadratic forms of an uneven number of indeterminates. Again, when m is a divisor of n, let n = hm, and let h, c, d^, ds represent the greatest numerical divisors of 5, C, F^, Fs respectively ; we find h-(^V (h\-(d'V ^_/^3\' C-zi^A* Art. 106.] BEPORT ON THE THEORY OP NUMBERS. 231 The first two of these equations show that the invariants of the three forms Fi, F^, Fs are so related, to one another, that we may imagine them to have been all derived by transformation from one and the same form (see Art. 80) ; the last two (which, it is to be observed, present an ambiguity of sign when — is even) show that the forms B and FJ', C and F^, are respectively identical, if we omit a numerical factor. Lastly, let $i, <I>2, $3 be any corresponding covariants of F^, F^, F^. The relation of covariance gives rise to the equations ttip — <i ^i{p(h,x^,...,x„)y.B " =$2(3/1.2/2, •••, 2/n)x^3*(2l>---,Z„), mp—q *i (a^i, 3^2, •••, «») X C^^ =$3 (zi, Z2, ..., z„) X F2* (yi, ..., y„), where p and q are the orders of the covariants in the coefficients and in the indeterminates respectively. Combining with these equations the values of 1 1 B and C already given, we see that $2 x -^s" and $3 x F.p are identical, ex- cepting a numerical factor ; i.e. that $2 a-rid $3 are either identically zero, or else numerical multiples of powers of F^ and F:^. If therefore two forms can be combined by multiplication so as to produce a third form transformable into their product, their covariants are all either identically zero or else are powers of the forms themselves. There is, consequently, no general theory of com- position for any forms other than quadratic forms, because aU other sorts of forms have covariants which cannot be supposed equal to zero, or to a multiple of a power of the form itself, without particularizing the nature of the form. And even as regards quadratic forms, we may infer that composition is possible only in cases of continually increasing particularity, as the number of indeter- minates increases. 106. Composition of Quadratic Fm^ms. — Preliminary Lemmas. — The follow- ing lemma is given by Gauss as a preliminary to the theory of the composition of binary quadratic forms (Disq. Arith., Art. 234) : — (L) ' K the two matrices and be connected by the equation A A-i A2 .. A. B BiB,...B^ a h = Oi a^ ... a^ &i 62 ... h^ 1 A B -" h i 232 REPORT ON THE THEORY OP NUMBERS. [Art. 107. in which the sign of equality refers to corresponding determinants in the two admit of no common divisor beside matrices ; and if the determinants of unity ; the equation A B in which the sign of equaUty refers to corresponding constituents in the two matrices, is always satisfied by a matrix | ^ | of the type 2x2, of which the determinant is h, and the constituents integral numbers.' * The subsequent analysis of Gauss can be much abbreviated if to this lemma we add three others. In their enunciations we represent by X, Y, x, y, four functions, homo- geneous and linear in respect of each of the n binary sets, ^j j/j, ^2 "/z. ••• > fn in J by A B and the matrices composed of the coefficients of X, Y and x, y, respectively ; by (P, Q, R), {P', Qf, R) quadratic forms of which the coefficients are any quantities whatever ; and by h an integral number. (ii.) * If X, Y, X, y, satisfy the n equations included in the formula dX dY dX dY _ ,/dx dy dx dy\ d^i drii dm d^i ^d^idm dm d^/ A B the matrices and satisfy the equation A B k (iii.) ' The greatest numerical common divisor of the n resultants dXdY dXdY A B d^i dm dm d^i is equal to the greatest common divisor of the determinants of (iv.) ' If the n resultants of X and Y be not all identically equal to zero, the equation px' + 2QXY+RY' = rX' +2Q'XY+E'y' implies the equations P = P', Q=Qf, R = R' 107. Gauss's Six Conclusions. — Talcing F, f, f to represent the forms {A, B, C) {X, Y)\ (a, 6, c) {x, yf, (a', &', c') (x, yj, of which the determinants are * For a generalisation of this theorem, see a paper by M. Baziu, in Liouville, vol. xix. p. 209 ; or Phil. Trans., vol. cli. p. 295. Art. 107.] REPORT ON THE THEORY OF NUMBERS. 233 D, d, d' ; let also M, m, m' be the greatest common divisors of A, 2B,C, of a, 2h, c, and of a', 26', c ; 01, m, m' the greatest common divisors of A, B, C, of a, h, c, and of a, V, c', respectively. Supposing that F is transformed into f y. f by the substitution X=paXx'-\-pi xy + ^2 x'y + p^ yy, Y=q^ xaf + q^ xy' + q^ x'y + q^ yy\ let us represent the two resultants dX dY dX dY , dX dY dX dY dx dy dy dx dx' dy dy dx by A and A' ; the six determinants of the matrix (taken in their i>o, Px, Pi, Ps qo, qi, q2, qa natural order) by P, Q, R, S, T, U ; the greatest common divisor of these six numbers by k, and the greatest numerical divisors of A and A' by B and 5' respectively, so that (Lemma 3) k is the greatest common divisor of S and h'. From the invariant property of the determinants of i^,/ and/' we infer A'** = ~f\ A2 = ^f\ D ^'2 = d'm\ DS' = dm\ Hence the quotients yr and •=- are squares. (Gauss's 1st conclusion.) Also D divides d'm^ and dm'^. (Gauss's 2nd conclusion.) But k is the greatest com- mon divisor of 5 and ^ ; therefore Dk'^ is the greatest common divisor of d'm'' and dm'. (Grauss's 4th conclusion.) Let y- = n', -jr = n', and let the signs of n and n' be so taken that A' = n'f, A = nf" ; these two equations are equi- valent to the six following : — P ^ R-S ^ ^ ^ '. ^ ^+^ ^ a "' = -r = n. (Q) 26 c 'a' 26' c (Gauss's 3rd conclusion.) Multiplying together the two resultants A and A', we obtain an identity, which we shall write at full : [iPoqi-Piqo)x'+{poqz-p3qo+P2qi-piq2)xy + (p^q3-p3q2)y''] ^ [{po qi -Pi qo) ^'^ + {po qz -pz qo +pi q^ -pt qi) xy + (pi q^ -p^ q^ y'^] = (2i qi - qo qz) {Po ^^ + Px ^y + B x'y + p^ yyy ■^{qoPz-\-Poqz-qiP2-qiPi){PoXX+piXy' + p^xy + p^yy) .... (I) X {q^ XX + q^ xy' + q^ x'y -I- q^ yy) + (PiP2- PoPa) {qo xx +qixy'+ q^ x'y + q^ yy'f. Hh 234 REPORT ON THE THEORY OF KUMBERS. [Art. 108. Comparing this identity with the equation AA' = nnff = nn'F, we find by Lemma 4 qxqi-qoqz qoP3+Poqs-qiP2-q2Pi _ PiPi-poP^ _ ^, fry^ The 5th and 6th conclusions relate to the order of the form compounded of two given forms. The equation AX' + 2BXY+ CY' = {ax^ + 2hxy + cy^) x {a'x" + 2b'x'y'+ c'y") shows that M divides mm'. But also mm' divides M¥. For operating on the d'^ d^ d''" equation just written with j-^ , , _, , -r-j successively, we find ,dX' ^^dXdY ^dY' ^ , ^r^dXdX ^/dXdY dX dY\ d^ dY■^^^ ^^^^^, L dx dy \dx dy dy dx / dx dy J~ ' ' • \J ) .dX\^j,dXdY^^dY'_^ , , A -^r-r + 2 XJ -T ; — h G -y-r = 0, mod mm . dy^ dy dy dy- Whence AA^, 2BA^, CA% and consequently MS^, are congruous to zero, mod mm'. Similarly MS'^ = 0, modm??i'; i.e. mm' divides Mk^. If then ^=1, i.e. if F he compounded offandf, M=m,m'. (Gauss's 5th conclusion.) Again, if in the congruences (y) we take m'm as modulus instead of mm' , we may omit the factor 2 in the second congruence, and may infer that AA^, £A2, CA2 are all divisible by m'm, i.e. that mm' divides iWi^ or Jtt, when F is compounded of/ and /'. It is also readily seen that iH divides mm' and mm'; whence observing that m = m or \m, m' = m' or \m , jltt = ilf, or \M, according asf,f', and F are derived from properly or improperly primitive forms, we conclude that if f and f he hoth derived from properly primitive forms, the foi'm compounded of them is also derived from a pi'operly primitive form ; hut if either f or f be derived from an improperly p^nmitive form, the form coin- pounded of them is derived from a similar form. (Grauss's 6th conclusion.) In the transformation of F into /x/', the form / is said to be taken directly or inversely, according as the fraction n is positive or negative. And similarly for /' and n. 108. Solution of the Prohlem of Composition. — It appears from the identity (I) that \i A,B, C, po,pi, p2>Ps> S'o. ?!> S'z* S's* be integral numbers satisfying the nine equations (Q), the form {A, B, C) {X, Yy will be transformed into the pro- duct of the two forms (a, h, c) (x, yY and (a', h', c) {x', y')'^ by the substitution X =po XX +pi xy' +P2 yx' -^p^ yy, Y=q^xx' + qi xy' + q^ yx + 5-3 yy'. Art. 108.] REPORT ON THE THEORY OP NUMBERS. 235 are to admit In order, therefore, to find a form, F, compounded directly or inversely of two given forms of which the determinants are to one another as two squares, we have to find eleven integral and two fractional numbers, satisfying the equations (Q) and (fl'), in which a, b, c, a, b', c, and the signs of n and n, are alone given ; the numbers Po, Pi, Pz, Ps, 5'o. ?i. 5'2) S'a. being further subject to the condition that the determinants of the matrix -^<" -^^i' -^^^i i 3 qo, qu q2, Ss of no common divisor. To determine n and n, we observe that the six deter- minants satisfy the identical relation PU— QT+ItS = 0; from which we infer, first, that P,Q,R-S,R + S,T,U must be relatively prime, if P, Q, R, S, T, U are to be so ; and secondly, substituting for the determinants their values given by the first six of the equations {Q), that dn- = d'lri^. Denoting by h' and S the greatest common divisors of P, R — S, TJ and of Q, R-\- S, T, so that 8 and ^ are relatively prime, we have evidently n'= + — , n= + — 7; the positive or negative signs being taken according as /' and f enter the com- position directly or inversely ; and the absolute values of 8 and iT being deter- mined by the equation ^^ d'm? = ^^ dm"^. The fractions n and n being thus ascertained, the values of P, Q, R, S, T, U are known from the equations (Q) : these values are all integral : for P, Q, R — S, R + S, T, U, this is evident from the equations {Q), and may be proved for R and S by means of the identity PU— QT+RS=0. We have next to assign such values to the constituents of the matrix 1 <" r^' P^' J^^ ^jj^t its determinants may acquire the known qo, qi, q2, q^ values of P, Q, R, S, T, U. To do so, it is sufficient * to obtain a fundamental set of solutions of the indeterminate system, x^U-X2 T+x^ S=0, 1 — XgU -\-X2R — x^Q = Q, -XgS+Xl Q — x^P + x^P = 0, = 0, (S) which is equivalent to only two independent equations. From the skew sym- * For a solution of the general problem, ' To find all the matrices of a given type, of which the determinants have given values,' see a paper by M. Bazin, in Liouville, vol. xvi. p. 145; or Phil. Trans., vol. cli. p. 302. For the definition of a fundamental set of solutions of an indeterminate system, see ibid. p. 297. It may be observed that the analysis of Gauss, which is exhibited in the text, is applicable to any matrix of the type n x {n + 2). Hh2 (2) 236 REPORT ON THE THEORY OF NUMBERa [Art. 108- metrical form of the matrix of this system, it appears that if 6g, 6^, 62, &3 be any multipliers whatever, any four numbers (x,, x^, X2, x^) proportional to -e,p +628+0,1, -6oQ-e,S +6,U, — OqR — QiT—Q^U , will satisfy the system (S), and in addition the equation ^0 a^o + ^1 aa + ^2 352 + ^3 a;3 = 0. Assigning, then, to 0o. ^1. ^2. ^3 any arbitrary values whatever, let jo) ^n g'2. ?3 be four numbers relatively prime, and proportional to the four numbers (2); let also 'Toqo + Triqi + Tr2q2 + irsq3=l; and employing Xo, xj, 7r2, Xg in the place of Oo, 61, 02, ^3, let us represent by po, Pi, Pi, Pa the solution of (S) thus obtained. We have thus two solutions of (S), satisfying respectively the relations ■^oPo + -^iPi + '^2P2 + -^3P3 = 0, and -Toqo + Triqi + ir^qi + Trsqs^l, which prove that the two solutions form a fundamental set, i.e. that the determinants = [P,Q,R,S,T,U]. Po,Pl,P2,P3 qo,qi,q2,q3 It only remains to show that the values of A, B, C, which are now sup- plied by the equations (Q'), are integral. Operating on the identity (I) with d^ d^ d^ d^ d^ d^ d^' d^' df' ^"^ ^^"° ^^*^ ^^' d^" %^' "^^ ^^' ^y reasoning similar to that which we have employed to establish the 5th conclusion, that 2 Ann, 2Bnn', 2Cnn, which are certainly integral numbers, are divisible by 2SS' if — ^ — and — ^ — are both even, and by SS' if either of these numbers is imeven. In the former case A, B, C are evidently integral ; in the latter, .^, 26 26' . . . , either — or — y is uneven, i.e. either m or m is. even, and the quotients of mm ■ ^ 2 A 2B 2C 2 Ann, 2Bnn, 2Cnn divided by SS' are ;, -,, ,; whence, again, A n r^ . , 1 * '>nm mm mm > -& , A, B, C are mtegral.* * Gauss shows that A, B, C are integral by substituting the values of j9j, ..., q^, ... in and observing that the results, after division by nn, are integral. The values of p„, ... are always obtained free from any common divisor by the process in the text; but Gauss has to determine four new multipliers, 5„ 6,, 0^, 0,, to obtain from the formulae (2) the exact values of q^,..., and not equimultiples of those values. M. Schlafli (Crelle, vol. Ivii. p. 170) has shown that Gauss's demonstration is coimected with a remarkable symbolical formula. Art. 109.] REPORT ON THE THEORY OF NUMBERS. 237 109. Composition of several Forms. — It will now be convenient to extend the definition of composition to the case in which more than two forms are compounded. K a quadratic form, F, be changed by a substitution, linear in respect of n binary sets, into the product of n quadratic forms, fi,f2, ...,fn, so that i = n F{X,Y)^ n {ciiXi^ + lbXiyi + CiyP^), i = l we shall say that F is transformable into /i x/2 x . . . x/„ ; and if the deter- minants of the matrix of the transformation are relatively prime, we shall say that F is compounded oi fi,f2, ...,/„. We shall retain, with an obvious extension, the notation of Art. 107. The invariant property of the deter- minant of F supplies the n equations from which we infer (1) that D, d^, d^, ... are to one another as square numbers, (2) that Dk^ is the greatest common divisor of the n numbers — '—Hm^. Ac- cording as the equation A,yj= A/yy-ny is satisfied by a positive or negative value of the radical, we shall say that f{ is taken directly or inversely. Adopting this definition, we can enunciate the theorem — 'If Fhe compounded oi'/iff, ...,/„, and F' be transformable into Jl ^J2 ^ • • • ^ /n > the forms being similarly taken in each case, F' contains F.' For we infer from (2) that D'k'^ = D, whence A',. = ^'A,., or by the Lemmas 2 and 1 of Art. 107, X' = aX + l3Y, Y' = yX+SY, «> P> y, ^ denoting integral numbers which satisfy the equation. a^ — /37 = F. We thus obtain the equation r{aX+pY, yX+SY) = F{X,Y), a,/3 7, ^ whence, by Lemma 4, F' is transformed into F by If i^ be compounded oi fi,/^, •••,/„, and a single transformation of F into yi x^ X ... xf„ be given, we may, by the same principles, find all the transforma- tions of i^ into the product of fi,/^, ...,f„, taken as in the given transformation. For if F{XQ,Y^ = Tlf represent the given transformation, and F{X,Y) = Tlf be any other transformation, we find X=aX, + ^Y„ Y=yX,^-8Y„ aS-Py=+l, 238 REPORT ON THE THEORY OF NUMBERS. [Art. 109. and consequently F{aX^ + ^ Y^, yX^ + SY^ = F (X, , Y^) ; a is, by Lemma 4, a proper automorphic of F. The formula is an automorphic of F, will therefore represent all the trans- or ill which «,i8 7, <5 formations required. If jP be transformable into/i x^^ x ... x/,, and <I> contain F, while /i, /a,...,/, contain 0i, ^2. •••. <^n» "^ will be transformable into (p^XipiX ... X(p„. This follows from a preceding general observation (Art. 105) ; but we must add here that if T, t,- denote any positive or negative units, according as the transformations of $ into F, and /( into <pi are proper or improper, while v,- denotes a positive or negative unit according as ff is taken directly or inversely, (pi will be taken directly or inversely according as Txr^xVi is positive or negative. This is apparent if we observe that the sign of the quantity -ff^J is altered by an improper transformation of X, Y, or ic,-, y,-, but is not altered by a transformation of any of the other sets. The theorem that ' forms compounded of equivalent forms, similarly taken, are themselves equivalent' is included in the preceding. We may, therefore, speak of the class compounded of any number of given classes. It is an important and not a self-evident proposition, that if jP be com- pounded of <p,f3,ft, ...,fn, and (p be compounded of ^^,^2, F is compounded of fiffi, ■••>fn- Let ^ = a^2 ^ 2 /8^j; + 7i;2, let fi be the greatest common divisor of a, 2/3, 7, and V the determinant of <f> ; let also X, Y transform F into <P x/3 x/4 X ••• x/„. Writing in X and Y for ^ and rj the bipartite expressions linear in x^y^, x^tfi, by which (p is transformed into fi xf^, we obtain a transformation of F into fi xfi X ... x/„. If ^ be the greatest common divisor of the determinants of the matrix of this transformation, Dk"^ is the greatest common divisor of the n numbers — ^ 11 m^. But this common divisor is the same as the greatest common i = n divisor of V x 11 m,-^, and the 71 — 2 numbers (l.fji'is = n -r-r- n m/, * = 3, ...,n; m^ ,=3 because V is the greatest common divisor of d^rn^ and dim^ (4th conclusion). Art. 110.] REPOBT ON THE THEORY OF NUMBERS. 239 and because fi = mim2 (5th conclusion) ; i. e., Dlc^ = D, or k"^ = 1, and F is com- pounded of fi,f2, ...,/„. Also, if i> 2,ff is similarly taken in both compositions, for A./. „.^ A,/. /1X/2X...X/, are identical ; and if ^ = 1, or 2, and *~ dxi dyf dX dY ^ ,dX dY di/i dxf \cZ| d>i <^x/3X...x/„ dX dY\ /d^ di dt] d^' \dxi dyi d^ dr, ;). dyi dXi' whence, if Q and tu,- be positive or negative units, according as <p andy) are taken directly or inversely in the composition of F and <p respectively, fi will be taken directly or inversely in the composition of F, according as O x to,- is positive or negative. By this theorem, the problem of finding a form compounded of any number of given forms is reduced to the problem of finding a form compounded of two given forms. For '\f f^, f^^, . . . , f^ be the given forms, we may compound the first with the second, the resulting form with the third, and so on until we have gone through all the forms, when the form finally obtained will be com- pounded of the given forms, as will immediately appear from successive appli- cations of the preceding theorem. We also see that we may compound the forms in any order that we please, or we may divide them into sets in any way we please, and compounding first the forms of each set, afterwards compound the resulting forms. If any of the given forms are to be taken inversely, we may substitute for them their opposites (Art. 92) taken directly. We may thus, without any loss of generaUty, and with some gain in point of simplicity, avoid the consideration of inverse composition altogether ; and, for the future, when we speak of the form compounded of given forms, or the class compounded of given classes, we shall understand the form or class compounded directly of the given forms or classes. 110. The solution of the problem of composition given in Art. 108 may be put into a form better suited to actual computation. The system (S) is evidently satisfied by [0, P, Q, R], and also by [-P, 0, —S, —Tl; and these solutions are independent, because the determi- nants of their matrix cannot all be zero unless P = 0, a supposition which may be rejected as it implies that a = 0, i.e. that cZ is a square. From this set of independent solutions a set of fundamental solutions is deduced, as follows. Let n be the greatest common divisor of P, Q, R; and let k be determined by the congruences q M ->S=0, R f- -T=0, mod /« 240 REPORT ON THE THEORY OF NUMBERS. [Art 110. O Tf which are simultaneously possible, because — and — have no common divisor with the modulus, while the determinant -(RS-QT)=^U- is divisible by it. The solutions are then a ftindamental set, and may be taken for [potPuPziPi], [qo, <lu<li, Js]* respectively. We thus find Ann' = ^, or ^=^'; 2Bnn' = R + S-2h^' P O Tf Multiplying this equation by — , — , — in succession, and attending to the congruences satisfied by k, we obtain the congruences P -r, ab' Q n ab R „ bb' + Dnn , > — B = — , —B = — , —B= , mod .4 ; ft. H fi fi fi fi P O If which determine B for the modulus A, because — , — , — are relatively prime. fi fi fi These determinations [viz. of A, and of B, mod A] are sufiicient for our purpose ; because, i£ B' = B + XA, the forms (A, B, ~-j—) and (A, R, ^ ) are equivalent. To obtain, therefore, the form compounded of two given forms (a, 6, c), {a, V, c'), we first take the greatest common divisor of d' m? and d m'* for D (giving to D the sign of d or d') ; we then determine n and n by the equations w = v/-yy, n' = \/-jr, and, representing by /* the greatest common divisor of an', an, bn' + h'n, we obtain A, B, C, from the system A = aa an' B = a¥ y fi a'n B = a'b fi bn'+Vn B = bb' + Dnn fi fi ■ mod A. Art. 111.] REPORT ON THE THEORY OF NUMBERS. 241 These formulae, which are applicable to every case of composition, and are therefore more general than the analogous formulae given by Gauss (Disq. Arith., Art. 243), are due to M. Amdt*, who has also given an independent investigation of them, though our limits have compelled us here to deduce them from Gauss's general solution of the problem of composition. That {A, B, C) is transformed into (a, h, c) x (a, b', c) by the substitution 1 ^ , b'-Bn , b-Bn , bb' + Dnn -BCbn' + b'n) , fx a ^ a ^ aa ^^ fj.Y= an'xy' + anx'y + (b'n + bn) yy, may be inferred from the values oi p^, ..., q^, ■••; or may be verified directly by observing that fji\_AX+{B + VD) F] = [ax + (& + w x/D) 3/] x [a'x' + (6' + n'vD) y']. 111. Composition of Forms — Method of Dirichlet. — Lejeune Dirichlet, in an academic dissertation (' De formarum binariarum secundi gradus compositione,' Crelle, vol. xlvii. p. 155), has deduced the theory of the composition of forms from that of the representation of numbers. The principles of this method are applicable to any case of composition ; but Dirichlet has restricted his investigation to properly primitive forms of the same determinant D. Let (a, &, c), {a, b', c) be two such forms ; let M and M' be two numbers prime to 2 D, and capable of the primitive representations M = am^ + 2 bmn + cn^, M' — a'm'^ + 2 b'm'n + c'n'^, by the forms (a, b, c) and (a', b' , c) respectively ; also let these representations appertain to the values w and w' of \/D, so that to' = D, mod M, w'-^ = D, mod M', * Crelle's Journal, vol. Ivi. p. 64. In the new edition of the Disq. Arith. (Gottingen, 1863), a MS. note of Gauss is printed at p. 263, containing the congruences by which B is determined in the case of the direct composition of two forms of the same determinant. The account of the theory of composition in the preceding Articles (106-109) differs from that in the Disq. Arith. (Arts. 234-243) chiefly in the use which is here made of the invariant property of the determinant. A different mode of treatment of Gauss's analysis is adopted by M. Bazin, in Liouville, vol. xvi. p. 161. In Arts. 108 and 110 we have endeavoured to supply the analysis of a problem which Gauss, as is not unusual with him, has treated in a purely synthetical manner (Disq. Arith., Arts. 236 and 242, 243) ; and it is for this reason that we have introduced the consideration oi fundamental sets of solutions of indeterminate systems, which are not explicitly mentioned in the Disq. Arith. It is perhaps singular that Gauss does not employ the identity PU—QT+ES= ; it was first given by M. Poullet DeUsle, in a note on Art. 235 in his Translation of the Disq. Arith. I i 242 REPORT ON THE THEORY OF NUMBERS, [Art. 111. and so that the forms (a, h, c), (a', h', c) are respectively equivalent to the forms If the values « and w' are concordant, i. e. if it is possible to find a number Q satisfying the three congruences Q' = D, mod MM', = «, mod M, Q = w', mod M\ (in which case the solution O of the congruence Q^=D,raodi MM', may be said to comprehend the solutions w and w of the congruences w^ = D, mod M, and w^ = D, mod M',) the form (mm; q. ^) wiU be a properly primitive form of determinant D, and will belong to one and the same class (which may be termed the class compounded of the classes containing (a, h, c) and {a', b', c')) whatever two numbers (subject to the con- ditions prescribed) are taken for M and M'. To prove this, a few preliminary remarks are necessary. (1) If the solutions w and w' are concordant, there is but one solution Q (incongruous mod MM') comprehending them. (2) The necessary and sufficient condition for the concordance of m and w' is w=u>', for every prime modulus dividing both M and M'. (3) If Q, m, w satisfy the congruence x^ = D for the modules MM', M, and M' respectively; and if, besides, = w, Q = ft)', for every prime divisor of M and M' respectively, w and w are concordant, and Q is the solution comprehending them. (4) The value of \/D, to which any given primitive representation (such as M=anC + 2hmn-\-cn'') appertains, may be defined by congruences, without employing the numbers IM and V which satisfy the equation mv — nij. = l (see Art. 86) ; in fact, we find am + {h + w)n = 0, mod M, {b — w)m + cn = 0, mod M ; whence also « = — 6, mod d, «»=+&, mod d', if d and d' are conamon divisors of M and m and of M and n. We may suppose that the given forms (a, b, c) and {a, ¥, c") are so prepared* * It is readily proved that a properly primitive form can represent numbers prime to any given number ; thus a form can always be found equivalent to a given properly primitive form, and having its first coefficient prime to a given number. This transformation will be frequently employed in the sequel: in the present instance, we have only to substitute for the given forms any two forms respectively equivalent to them and having their first coefficients relatively prime. Art. 111.] REPORT ON THE THEORY OF NUMBERS. 243 that the representations of a and a by them appertain to concordant values of VD ; i. e. that we can find a number B satisfying the congruences B^ = D, mod aa, B = h, mod a, B = h', mod a'. Let -, — = C; the forms (a, h, c), (a, h', c) are then equivalent to (a, B, a'C), (a, B, aC) respectively; and if X = XX — Cyy , Y = axy' + ax'y + 2 Byy', we find by actual multiplication oa'X'' + IBXY^ GY^ = {ax'' + 2Bxy + aCy") x {ax^ + 2Bxy' + aCx% From this equation (which is included as a particular case in the formulae of M. Amdt) it appears that MM' is capable of representation by {cm, B, (J) ; it can also be shown (1) that this representation is primitive ; (2) that it appertains to a value of \/D, mod MM', comprehending the values w and w, to which the representations of M and M' by (a, h, c) and {a, h', c) respectively appertain. (1) If x, y, x, y, and X, Y are the values of the indeterminates in the representations of M, M', and MM' by (a, B, a'C), (a', B, aC), and (aa, B, C), the hypothesis that X and Y admit of a common prime divisor p is expressed by the simultaneous congruences xx' — Cyy' = 0, axy' + a'x'y + 2 Byy = 0, mod -p. These congruences are linear in respect of the relatively prime numbers x' and y ; their coexistence implies, therefore, that p di^ddes their determinant M ; similarly it may be shown that p divides M' ; so that <o = w', mod p, because (0 and 0)' are concordant. The congruences satisfied by w and w' now give the relations ax^■(B^w)y = ^, aV + (5 + «)y'=0, mod j); whence, eliminating x and x' from the congruence F=0, and observing that 2(o is prime to M, and therefore to p, we find yy'=0, modjp. If 2/ is divisible hj p, we infer, from the congruence X = Q, that x' is also divisible by p; but the* congruences satisfied by (jd and «' give in this case the contradictory results u)=+B, u)= —B; i.e. y is not divisible by p, and similarly it may be shown that y is not divisible by p. The congruence yy'=0, mod p, is therefore im- possible ; or the representation of MM' by {cm, B, C) is primitive. (2) Let Q' be the value of \/D, to which this representation appertains ; and let p be any divisor of M ; then Q.' satisfies the congruences aa'Z+ (B-l-Q')r=0, {B-Q')X+CY=0,modip; and it will be found, on substituting the values of X and Y, that these con- I i 2 244 REPORT ON THE THEORY OF NUMBERS. [ah. 112. gruences are also satisfied by w ; whence it follows, since either X or F is prime to p, that 0' = o), mod p. Similarly, if ^J be a prime divisor of M', Q'=u, mod p ; or Q' is a solution of the congruence Q^=D, mod MM', comprehending the solutions u) and «'. Hence Q'= Q, mod MM', and the form (J/Jf-.Q,^) is equivalent to (cut, B, C), because either of them is equivalent to (MM-.or,^). The equivalence of all the forms included in the expression (MM-. 9. 5j?) is therefore demonstrated. It will be seen that Dirichlet's method may be applied to the composition of any number of forms, and that the theorems of Art. 109 present themselves as immediate consequences of his definition of composition. 112. Composition of Classes of the same Determinant. — We shall now con- sider more particularly the composition of classes of the same determinant D. We represent these classes by the letters /, (p, ..., and we use the signs of equality and of multiplication to denote equivalence and composition respec- tively*. The following theorems are then immediately deducible from the six conclusions of Art. 107, and from the formulae of Art. 110. (i.) ' If/ be a properly primitive class, /x $ is of the same order as *.' (ii.) 'A class is unchanged by composition with the principal class.' In consequence of this property, it is sometimes convenient to represent the principal class by 1. (iii.) ' The composition of two opposite t properly primitive classes produces the principal class.' If, then, / denote any properly primitive class, we may denote its opposite by/~"\ and we may write /x/~^ = 1. * * Gauss uses the sign of addition instead of that of multiplication ; thus /x <^ is /+ (^ in the Disq. Arith., and /" is nf. The change appears to have been introduced by his French translator, and to have been acquiesced in by subsequent writers. t Two classes which are improperly equivalent are called opposite, because they contain opposite forms (see Art. 92). Art. 112.] REPORT ON THE THEORY OF NUMBERS. 245 (iv.) 'If / be a given properly primitive class, and <I> any given class, the equation Fxf=^ is always satisfied by one class, F, and by one only; viz. by the class i^=*x/-i.' (v.) ' If <I>i, <I>2, ... be all different classes, and/ be a properly primitive class, fx ^i,fx ^2i ■■• are aU different classes.' (vi.) 'A properly primitive ambiguous class produces by its duplication the principal class ; ' for an ambiguous class is its own opposite. Conversely, if <f>^ = l, i.e. a (f) be a class which, by its duplication, produces the principal class, is a properly primitive ambiguous class; for we find ^^x^~' = ^~^, whence (f> = (p~ \ or (f> and its opposite are properly equivalent. (vii.) ' The class compounded of the opposites of two or more forms is the opposite of the class compounded of those forms.' It follows from this, or from (vi.), that a class compounded of ambiguous classes is itself ambiguous. (viii.) Let $o, <I>,, ..., $<„_i represent all the classes of det. D, and of a given order 12; and let 1, fi, fi, ■■■, f„-i represent the properly primitive classes of the same determinant ; it may then be shown that w is a divisor of n, and that, given two classes of the order Q, there always exist - properly primitive classes, which, compounded with one of them, produce the other. Assuming, for a moment, that a form $o exists, such that the w equations included in the formula ^oxf=^i^ can all be satisfied, we see that each of these equations is satisfied by the same number of properly primitive classes /; for if the equation ^(,xf=^„ be satisfied by k primitive classes, 1, ^j, ^2> ••■» •^'ft-u t-he equation ^o^f—^fi, which is, by hypothesis, satisfied by a single class,/,, is also satisfied by the ^ — 1 classes /, x ^i, ...,/tX^ft_i, but by no other class. Since, then, the classes (poxf, of which the number is n, represent every class of the order Qk times, we have evidently n = kw. It is also readily seen that every equation of the type 4>yX/=$^ admits of k solutions; and thus it only remains to justify the assumption on which the preceding proof depends. If the order be derived by the multiplier m from a properly primitive class of determinant A = — , we may take for 'I'o the class represented by the form (m, 0, — A«i) ; if Q be derived from an improperly primitive class, we take for ^0 the class represented by the form (2m, m, -^m (A - 1)). Representing fl>^ in the first case by the form {ma, mb, mc), and in the second by the form (2 ma, mh, 2mc), and supposing (as we may do) that a in each case is prime 246 REPORT ON THE THEORY OF NUMBERS. [aH. 113. to 2D, we see that the forms (a, mb, mJ^c) and (a, hm, icm^) are properly primitive ; and we find by the formulae of composition (Art. 110), (m, 0, - Am) X (a, bm, cm'') = ( ma, mh, mc), {2m, m, - ^w(A - 1)) x {a, bm, ^cm^) = {2m<t, mb, 2mc) ; i.e. the equation $oXy=$^ can be satisfied for every value of /w, 113. Comparison of the numbers of Classes of different Orders. — To deter- mine the quotient - of the last Article, Gauss investigates the properly primitive classes of det. D, which, compounded with the classes (to, 0, — Ato) and (2to, m, — ^to(A-I)), reproduce those classes themselves. He thus employs the theory of composition to compare the number of properly primitive classes of a given determinant with the number of classes contained in any other order of the same deter- minant ; or, which comes to the same thing, to compare the numbers of classes, of any given orders, of two determinants which are to one another as square numbers (Disq. Arith., Arts. 253-256). We have already seen (Art. 103) that the infinitesimal analysis of Dirichlet supplies a complete solution of this pro- blem ; whereas, in the case of a positive determinant, the result in its simplest form was not obtained by Gauss. It has, however, been recently shown by M. Lipschitz (Crelle, vol. liii. p. 238) that the formulae of Dirichlet may be deduced, in a very elementary manner, from the theory of transformation. We propose in this place to give an account of this investigation, and to point out its relation to the method pursued by Gauss. We begin with the theorem — ' Every properly primitive class of determinant De^ is contained in one, and only one, properly primitive class of determinant D.' Let (A, B, C) be a properly primitive form of det. De^, in which A is prime to e ; let B" be determined by the congruence eR = B, mod A, and C by the B'^-D equation C = — ^ — ; then the forms {A, B, C) and {A, Re, C'e^) are equi- valent ; but {A, Re, C'e^) is contained in {A, B', C), therefore also {A, B, C) is contained in {A, R, C), that is, in a properly primitive form of determinant D. Again, if (a, b, c), {a', b', c) are two forms of det. D, each containing {A, B, C) these two forms are equivalent. For applying to {A, B, C) the m, k system of transformations of modulus e, included in the formula 0, fJi (Art. 88), Art. 113.] REPORT ON THE THEORY OF NUMBERS, 247 we readily find that, of the resulting forms, one, and only one, will have its coefficients divisible by e^ * ; therefore the class represented by (A, B, C) con- tains one, and only one, class of det. De*, and of the type {e^p, e'^q, eV). But, applying to {A, B, C) the transformations inverse to those by which (a, h, c) and {a, h', c) are changed into (A, B, C), then {A, B, C) is changed thereby into (e^a, e^b, e^c) and {e^a',e^h', e^c); these two forms are therefore equivalent; i.e. {a, h, c) and {a, b', c) are equivalent. We have next to ascertain how many different properly primitive classes of determinant Be^ are contained in the class represented by (a, b,c), a properly primitive form of det. D, in which a may be supposed prime to e. Applying to (a, b, c) a complete system of transformations of modulus e, we inquire, in the first place, how many of the resulting forms are properly primitive. For this purpose we observe that if e = eiXe2X ejX ..., {e^, e^,... representing factors of which no two have any common divisor), a complete system of transformations for the modulus e is obtained by compounding, in any definite order, the systems of transformations for the modules e^, 62, ...; i.e. if |ei|, | 62], ... be symbols representing complete systems of transformations for the modules gj, 62) •••, every transformation of modulus e is equivalent by post-multiplication f to one, and only one, of the transformations | Ci | x | gg | x | ^3 1 x It will, therefore, be sufficient to determine the number of properly primitive forms obtained by applying to a properly primitive form a complete system of transformations for a modulus which is the power of a prime. Let p be an uneven prime, and p'^-y, h let {a, b, c) be changed into {A, B, C) by 0. P' a formula which will repre- sent a complete system of transformations for the modulus p", if y receive every value from to a inclusive, and if ^ be the general term of a complete system If m, k 0,/x transform {A,B,C) into {P,Q,2i), we have P = Am', Q = 7n{Ak + Bi).), I{ = Ak'' + 2 Bkn+Cn''. Observing that A is prime to e, we infer from the congruence F = 0, mod e^ that wi = e, fi = 1 ; the congruence Q — 0, mod e^, then becomes Ak + B = 0, mod e, giving one, and only one, value of k, mod e ; and this value satisfies the remaining congruence i? = 0, mod e% since AE = {Ak + Bf — De'. t If I il I and I B | are two transformations connected by the symbolic equation |5| = l^|x|F|, in which | F | is a unit transformation, | A | and | B | are said to be equivalent by post-multiplication, or to belong to the same set. A complete system of transformations for any modulus contains one transformation belonging to each set. 248 REPORT ON THE THEORY OF NUMBERS. [Art. 113. of residues, mod p""'' ; we find A = ap«<'-T', B = (ak + hpi) p'-f, C=ak'^ + 2 hkpt + cp^"^ ; whence, if 7 = a, [A, B, C) is properly primitive ; and if so, or not, for every other value of 7, according as C is not, or is, divisible by p. If 7 = 0, we have 1 + ( — J values of k, incongrouous mod p"^ ; if 7 have any value intermediate between and a, we have C=0, for p'^-y-^ values of k, incongruous mod p"'"*. Hence the number of properly primitive foniis is [1 +2) +P'' +... +i?°] -j3°- ' [1 + (—)]- 0°- * +P'- »+... + 1] =^[-|(f)] = and similarly if ^ = 2 it will be found that the number of properly primitive forms is 2°. Hence the number N of properly primitive forms, arising from the application of a complete system of transformations of modulus e to the form {a, b, c), is e IT 1 ( — j , p denoting any uneven prime dividing e. It remains to determine the number of non-equivalent classes in which these N forms are contained. For brevity, we consider the case of a positive deter- minant. Let [T^, Z7J represent any solution of the equation T' — DU' = 1, and let o- be the index of the least solution of that equation which is also a solution of T^ — e'^DU^ = l, i.e. let a- be the index of the first number in the series £/,, U.2, ... which is divisible by e ; also let (A, B, C) represent any one of the N properly primitive forms into which (a, b, c) is transformed. The trans- formations of modulus e by which (a, b, c) is changed into {A, B, C) belong to a- different sets, the transformations of the same set being equivalent by post- a, /3 7, 3 multiplication, but those of difierent sets not being so equivalent. For if be a transformation of (a, b, c) into (A, B, C), any other transformation is repre- sented (Art. 89) by the formula and these two transformations will or wUl not belong to the same set, according X, fx as a unit transformation a, ^ 7, S , satisfying the equation a,fi \,/x A X X yj ". P aU,,T, + bU, a,/8 7, S Art. 113.] EEPOET ON THE THEORY OF NUMBERS. 249 does or does not exist, we find e X S, -13 -y, a Premultiplying each side of this equation by X,M ^ eT,-BU,, -CU, v,p AU,,eT, + BU, whence, observing that A, B, C are relatively prime, we see that A, ju, v, p are or are not integral according as U^ is, or is not, divisible by e ; a conclusion which implies that the transformations of (a, b, c) into (A, B, C) are contained in o- different sets. It thus appears that, of the N transformations which applied to (a, b, c) give properly primitive forms, there are o- which give forms equivalent to {A, B, C) ; i.e. the number of properly primitive classes of det. Z>e^, contained in {a, b, c), a properly primitive class of det. D, is ^=^nri-l(^)1; a result which is in accordance with the formula of Dirichlet (Art. 103). If D be negative, we have only to put o- = 1, as is sufficiently apparent from the preceding proof; if, however, D = — 1, <r = 2. The properly primitive classes of det. De^, into which a given properly primitive class (a, 6, c) of det. D is transformable, are always such that, com- pounded with the class (e, 0, -De), they produce the class (ea, eb, ec). For let (a, h, c) be transformable into (A, B, C) of det. De^, and let us take a form of the type {A, B'e, C'e% equivalent to {A, B, C) ; then {a, b, c) and {A, B\ C) are equivalent. But (e, 0, -De) X {A, B'e, CV) = (e^, eB', eC), therefore also (e, 0, — De) x {A, B, C) = {ea, eb, ec). And, conversely, the classes which, compounded with (e, 0, —De), produce (ea, eb, ec) are precisely the classes into which (a, b, c) is transformable. Thus the properly primitive classes of det. De'^, which, compounded with (e, 0, — De), reproduce that class itself, are no other than the properly primitive classes of det. De^ into which (1, 0, —D) is transformable. And it is by this substitution of a problem of transformation for a problem of composition that M. Lipschitz has simplified and completed the analysis of Gauss. A method similar in principle is applicable to the comparison of the num- bers of properly and improperly primitive classes. We can first show that if D = l, mod 4, the double of every properly primitive class of det. D arises by a transformation of modulus 2 from one, and only one, improperly primitive class of the same determinant ; viz. if (a, b, c) is a given properly primitive 250 REPOKT ON THE THEORY OF NUMBERS, [Art. 113. form*, in which a and 6 are uneven, (2a, b, \c) is improperly primitive, and is changed into (2 a, 2&, 2 c) by ' 0,2 and, again, if {2p, q, 2r), {2p\ q', 2r') are two improperly primitive forms, each of which is transformable into (2a, 26, 2 c) these two forms are equivalent, because (a, 6, c) is transformable into {ip, 2q,^r) and also into {^p', 2q', 4/), while it can be shown that (a, b, c) is transform- able into the double of only one improperly primitive class. Also, applying the system of transformations. 1,0 2,0 2,1 0,2 9 0,1 > 0,1 , to the improperly primitive form {2p, q, 2r), we obtain, if Z)=l, mod 8, the double of only one properly primitive form : in this case therefore the numbers of properly and improperly primitive classes are equal. J£ D = 5, mod 8, we obtain the doubles of three properly primitive forms ; and we have to decide to how many different classes «,i8 these three forms belong. It appears from Art. 89, that if 7, S be a trans- formation of {2p, q, 2r) into the double of a properly primitive form (a, b, c), all the transformations are included in the formula [T'j,, £/"J denoting any solution of the equation T^ — DU^ = i. Taking the case of a positive determinant, and employing the same reasoning as before, we infer that if Ua be the first of the numbers t/j, U2, ... which is even, these trans- formations are contained in a- different sets. But o- is either 1 or 3 according as Ui is even or uneven (see Art. 96, (vi.)) ; the three forms will therefore re- present three classes or one, according as f/i is even or uneven ; and the number of properly primitive classes, in these two cases respectively, wUl be three times the number of improperly primitive classes, or equal to it. If D be negative, the three forms will belong to different classes ; and there wUl be three times as many properly as improperly primitive classes. From this statement, how- ever, we must except the determinant —3, which has one properly and one improperly primitive class. It will be found that the properly primitive class or classes, into the double * I K (a, b, c) is properly primitive, a and c uneven, b even, transforms (a, b, c) into 1, ±1 + , 1 2 X an improperly primitive form ; i.e. when there are improperly primitive forms at all, which implies a+c = 0, mod 4. Either sign may be taken.} Art. 114.] REPORT ON THE THEORY OF NUMBERS. 251 of which a given improperly primitive class can be transformed, and which in turn can be transformed into the double of the given class, are also the class or classes which, compounded with the class (2, 0, —^{D- 1)), produce the given class. Thus every improperly primitive class is connected either with one or three properly primitive classes (see Art. 98, note, and Art. 118). 114. Composition of Genera. — Let /and/' be two properly primitive classes of det. D, m and m' two numbers prime to one another and to 2 D, and repre- sented by/ and/' respectively; then mm' is represented hj fxf. Hence the generic character of /x/' is obtained by multiplying together the values ol the particular characters of / and /'. For those generic characters which are expressed by quadratic symbols this is evident, since /nvm'\ /in\ /m'\ (-^) = (p ) ^ (7) ' and it is equally true for the supplementary characters, since it will be found that The genus F, in which /x/' is contained, is said to be compounded of the genera y and 7', in which / and /' are contained ; and this composition is expressed by the symbolic equation F = <y x 7'. It will be seen that the composition of any genus with itself gives the principal genus. The same considerations may be extended to improperly primitive classes. Thus, if/ and /' be respectively properly and improperly primitive, m and m uneven numbers prime to one another and to D, represented by / and \f', the genus of the improperly primitive class, /x/', may be inferred from the number mm, i.e. it is obtained by the composition of the generic characters of/ and/'. Or, again, if / and /' be both improperly primitive, so that the class compounded of them is the double of an improperly primitive class, the generic character of this improperly primitive class is obtained by compound- ing those of the two given classes. It follows, from these principles, that the number of classes in any two genera of the same order is the same. For if ^Jj, $2> •••> ^n ^^ ^^1 ^^^ classes of any genus of properly or improperly primitive forms, F^ a class belonging to any other genus of the same order, and (j> a properly prmaitive class 8atisf)ring the equation ^^xcjy-Fi, the classes <I>iX ^, ..., ^„ x ^ are all different, and all belong to the genus (F) ; consequently (F) has at least as many classes as ($), and vice versd ($) has at least as many as (F), i.e. they both contain the same number of classes. K k 2 252 REPORT ON THE THEORY OF NUMBERS. [aH. U5. 115. Determination of the Ntimher of Ambiguotis Classes, and Demon- stration of the Law of Quadratic Reciprocity. — The number of actually existing genera of properly primitive forms cannot exceed the number of properly primitive ambiguous classes. For let n be the number of classes in each genus, k the number of actually existing genera, so that kn is the number of properly primitive classes; let also 1, A^, A2, ..., Aj,_i be the properly primitive ambiguous classes. Every class produces, by its duplication, a class of the principal genus ; and if ^ be a class of the principal genus produced by the duplication of X, K is also produced by the duplication of Xy.A^,X-KAz,...,Xy.Aj,_i, but by the duplication of no other class. If, therefore, there be n classes in the principal genus which can be produced by duplication, the whole number of properly primitive classes is A x n, i. e. hn = hn. But n ^ n, therefore k < h. It may be inferred from Art. 112, (vii.), that all genera which contain any ambiguous classes contain an equal number of them. We shall immediately see that the number of ambiguous classes is equal to the number of genera, and is consequently a power of 2. The number of ambiguous classes in any genus is, therefore, either zero or a power of 2 ; and if any genus contain 2* k ambiguous classes, such classes will exist only in ^ genera. Gauss determines the number h of properly primitive ambiguous classes by very elementary reasoning. He first finds the number of properly primitive ambiguous forms of one or other of the two types {A, 0, C) and {2B, B, C), and then assigns the number of non-equivalent classes in which these forms are contained. Let D be divisible by n. different primes ; and let us except the case Z)= — 1. Resolving D in every possible manner into two positive or negative factors, having no common divisor but unity, we find 2''+i properly primitive forms of the type {A, 0, C) ; but we shall diminish this number by one-half by rejecting one of the two equivalent forms {A, 0, G) and (C, 0, A), viz. that in which [A] > [C]. There are no properly primitive forms of the type {2,B, B, C) unless D = 3, mod 4, or D = 0, mod 8; for one or other of these congruences is implied by the equation D = B(B — 2C), because C is uneven. Resolving D into any two factors relatively prime, if Z)=3, mod 4, and having 2 for their greatest common divisor, if X) = 0, mod 8, we take one of them for B, the other tor B — 2C; and we obtain, in either case, 2/*+^ properly primitive forms of the tyTpe {2B, B, C). If BB' = - D, it is easUy seen that Art. 115.] REPORT ON THE THEORY OP NUMBERS. 253 the forms {2B, B, C) and (25', B', C)* are eqiiivalent. We may thus diminish hy one-half the number of forms of the type (2JB, B, C), rejecting those in which [£]>\/[i)]. We conclude, therefore, that if we now denote by fi the number of uneven primes dividing D, we have in all 2'' + 2 ambiguous forms when D = 0, mod 8, 2^ when Z) = l, or =5, mod 8, and 2'' + i in every other case. These ambiguous forms we shall call 0, and we observe that their number is equal to the whole nimiber of assignable generic characters (Art. 98). To find the number of non-equivalent classes in which these forms are contained, we consider separately the case of a positive and of a negative determinant. For a negative determinant, we diminish by one-half the number of the forms by rejecting the negative forms. The remaining forms, if of the type (A, 0, (7), are evidently reduced, because A <C; if of the type {2B, B, C), they are also reduced, unless 2B> C, an inequality which implies that (C, C-B, C), to which {2B, B, C) is equivalent, is reduced (Art. 92). The number of [positive] ambiguous classes is, therefore, one-half the number of the ambiguous forms Q. For a positive determinant, we deduce from the forms Q an equal number of reduced ambiguous forms. Thus {A, 0, C) is equivalent to {A, TcA, C) ; and because [^] < \^D, this form is reduced, if hA be positive and be the greatest multiple of \_A~\ not surpassing \/D. Similarly (25, i2h + l)B, C) is equivalent to (25, B, C), and is reduced if (2^-1-1)5 be positive, and be the greatest uneven multiple of [5] not surpassing v'D. There are, therefore, as many reduced ambiguous forms as there are forms in 12 ; and there are no more, because it is readily seen that every reduced ambiguous form is included in one or other of the two series of forms {A, hA, C) and (25, (2^-f 1)5, C) which we have obtained. But every ambiguous class contains two reduced ambiguous forms (Art. 94) ; we infer, therefore, that for positive as well as for negative determinants the number of ambiguous classes is one-half the number of the forms Q, i.e. one-half of the number of assignable generic characters. Combining this result with the theorem at the commencement of this * When the first two coefficients of a form are given, the third is given also ; thus C is here B"^—D used for . Similar abbreviations will be employed occasionally in the sequel. The symbols \A\ &c. are used, as in Art. 92, to denote the absolute values of the quantities enclosed within the brackets. 254 REPORT ON THE THEORY OF NUMBERS. [Art. 116. article, we obtain a proof of the impossibility of at least one-half of the assignable generic characters. As this proof is independent of the law of quadratic reciprocity, we may employ the result to demonstrate that law. [Gauss's second demonstration, Disq. Arith., Art. 262.] Let p and q be two primes, and first let one of them, as p, be of the form 4n + l. If (— ) = —1, we infer that (—)=—!; for if (—)=+!, we should have 0)*=^, mod g', and consequently there would exist a form (a, w, —\ of det. p, of which the character would be (— )= —1. *-e. there would be 2 genera of forms of determinant p. Similarly, if (—)=+!, we have <«)''=+j, mod p; and (p, w, ( ^) is a form of det. +q^. K +5 be of the form 4n-|-l, there will be but one genus of forms, i.e. the principal genus; whence (— )= +1- These two conclusions are sufficient to establish the theorem of reciprocity when one of the two primes is of the form 4n + l. If both p and q be of the form 4% + 3, there are four assignable characters for the determinant pq. are possible, as is shown by the existence of the forms {1,0,-pq), (-1,0, 2)q); the other two are therefore impossible. Hence in the form [p, 0, — q) we must have either (^).,.(^), „, (f)--l-(f> which expresses the theorem of reciprocity for this case. The supplementary theorems relating to 2 and — 1 can be similarly proved. 116. Equality of the Number of Genera and of Ambiguous Classes. — In the preceding article it has only been shown that k cannot exceed h. But, as we have already seen (Art 102) that the number of actually existing genera is one-half the whole number of assignable generic characters, we know that k = h. To prove tliis, by the principles of the composition of forms, it is sufficient to show that n = n, i.e. that the problem 'to find a class which by its duplication shall produce a given class of the principal genus' is always resoluble. This problem Gauss actually solves (Disq. Arith., Arts. 286, 287) ; he shows, first, that any proposed binary form, belonging to the principal genus of its own determinant, can be represented by the ternary Art. 116.] REPORT ON THE THEORY OF NUMBERS. 255 quadratic form X' — 2 YZ ; and, secondly, that from this representation we can always deduce a binary form, which shall produce by its duplication the proposed form. This solution implies a previous investigation of the theory of ternary quadratic forms, and cannot be properly introduced here. A more elementary method, however, has been given by M. Amdt (CreUe, Ivi. p. 72). Let D = AS^, S^ representing any square dividing D ; M. Amdt observes that the ratio of the number of actually existing genera to the whole number of assignable generic characters is the same for each of the two determinants D and A. To prove this we make use of the following subsidiary proposition : — ' If /= (a, &, c) be a properly primitive form of any det. D, and if 8 if and 6 be two numbers relatively prime, the necessary and sufficient condition for the resolubility of the congruence ax^ + 2bxy + cy^ = e,mod8M (A) is that the supplementary characters of / (if any), and the particular characters of/ (if any) which relate to uneven primes dividing both M and D, should coincide with the corresponding characters of 6.' We may add (though this is not necessary for our present purpose), that if 01 and 02 he two values of 6 for each of which the congruence (A) is resoluble, it is resoluble for each an equal number of times. On reference to the Table in Art. 98, it will be seen that the particular characters proper to the determinant A are included among the particular characters proper to D. Let then (F) and (F, F') represent any two com- plete generic characters for the determinants A and D, the particular cha- racters common to the two complete characters having the same values attri- buted to them in each. It may then be shown that the genus (F, F') is or is not an existent genus, according as (F) is or is not existent.- For (1) if (F, F') be actually existent, let be a number prime to 2D and capable of primitive representation by some class of that genus ; the congruence w^ = D, mod Q is therefore resoluble ; i. e. the congruence w^ = A, mod Q, is resoluble, so that 6 can be represented by a class of properly primitive forms of det. A, or the genus (F) is actually existent. And (2) if (F) be an existing genus, let / be a form included in (F), and Q a nimiber prime to 2Z> and satisfying the generic character (F, V). It appears from the subsidiary proposition that some number O of the linear form SmD-^d is capable of representation by/; if ^ be the greatest common divisor of the indeterminates in the repre- sentation of 9 by/, the congruence ft>^ = A, and consequently the congruence 256 REPORT ON THE THEORY OF NUMBERS. [Art. 117. Q Q ta^ = D, Is resoluble for the modulus -zz] i.e. ^, the character of which coincides with the character of 6, and therefore with that of the genus (F, V), is capa- ble of representation by a form of det. D, or (F, F') is an actually existing genus. If, then, K be the number of particular characters contained in (F, F') and not in (F), the numbers of actually existing genera and assignable generic characters for the det. D are each 2* times the corresponding numbers for the det. A. It appears from this result that it will be sufficient for our present purpose to consider determinants not divisible by any square. If {a, 6, c) be a form of the principal genus of such a determinant (we suppose that a is prime to D), the equation ax^ + 2hxy + cy^ = w^ is resoluble with values of w prime to D ; for if a = aSi^, (5^ representing the greatest square divisor of a, the equation is certainly resoluble in relatively prime integers, by virtue of a celebrated theorem of Legendre * ; and the values of ^ which satisfy it are prime to D ; whence, if ^ — 6»; X. X = fi^——, y = firi, w = f.\, U fi denoting a multiplier, which renders the values of x, y and w integral and relatively prime, the equation ax^ + ^hxy + cy^^w^ will be satisfied, and the values of w will be prime to D. The form (a, h, c) is therefore equivalent to a form of the type {m^, X, v) ; and this form is produced by the duplication of (w, X, vw) if to be uneven, and of (2 w, X + w, /) if w be even. 117. Arrangement of the Classes of the Principal Genus. — If C be a class of the prmcipal genus, the classes C, C^, C", ... wUl all belong to that genus. And it will be found, by reasoning similar to that employed in Euler's second proof of Format's theorem (see Art. 10 of this Report), (1) that the classes 1, C, C^, ... are all different until we arrive at a class C^, equivalent to the principal class ; (2) that m is either equal to, or a divisor of, the number n of classes in the principal genus ; (3) that if (7'' = 1, r is a mul- tiple of ft. The jM classes C, C'\ C^, ...,C''~^, 1, are called the period f of the class C; C is said to appertain to the exponent /u ; and the determinant is * Th6orie des Nombres, ed. 3, vol. i. p. 4 1 ; Disq. Arith., Art. 294. t These periods of non-equivalent classes are not to be confounded with the periods of equivalent reduced forms of Art. 93. Art. 117.] REPORT ON THE THEORY OP NTTMBERS. 257 regular or irregular according as classes do or do not exist which appertain to the exponent n. With the former case we may compare the theory of the residues of powers for a prime modulus ; with the latter the same theory for a modulus composed of different primes (see Art. 77). (i.) When the determinant is regular, we may take any class appertaining to the exponent « as a basis, and may represent all the classes of the principal genus (to which we at present confine ourselves) as its powers. It will then appear (1) that if c^ be a divisor of n, the number qf classes appertaining to the exponent d \a ^{d); so that, for example, the number of classes that may be taken for a base is v|/- («) : (2) that if ef= n, the equation X* = 1 will be satisfied by e classes of the principal genus ; and if these classes be repre- sented by Ai, A2, .-.tAg, each of the equations X^ = A will be satisfied by / different classes of the same genus : (3) that the only classes of the prin- cipal genus which satisfy the equation X^ — 1 are those which satisfy the equation X^ = 1, where d is the greatest common divisor of h and n. It will be seen in particular that the equation X^ = \ admits of only one, or only two solutions, according as » is uneven or even ; ^. e. the principal genus of a regular determinant cannot contain more than two ambiguous classes. To obtain a class appertaining to the exponent n. Gauss employs the same method which serves to find a primitive root of a prime number (Art. 13 ; Disq. Arith., Arts. 73, 74), and which reposes on the observation, that if A and B be two classes appertaining to the exponents a and /3, neither of which divides the other, and if M, the least common multiple of a and |8, be re^ solved into two factors 'p and 5, relatively prime and such that -p divides o and 5 divides /3, the class A^ x jB' will appertain to the exponent M. (ii.) When the determinant is irregular, the classes of the principal genus cannot be represented by the simple formula C*, and we must employ an expression of the form Ci'i x Cg's x Cg's.... To obtain an expression thus representing all the classes of the principal genus, we take for Cj a class appertaining to the greatest exponent 6^ to which any class can appertain ; and in general for C^ we take a class appertaining to the greatest ex- ponent 6^ to which any class can appertain when its period contains no class, except the principal class, capable of representation by the formula Ci'i X C/i! X ...xC^.j'V n The number - = 02 x ^3 x • • • is called by Gauss the l1 258 REPORT ON THE THEORY OF NUMBERS. [Art. 118. n n exponent of irregularity ; and similarly we might term ^-^, , &c,, the second, third, &c., exponents of irregularity. From the mode in which the formula Ci'j x Cz's x ... is obtained, it can be inferred that 0i is divisible by 6^, 02 by 63, and so on ; whence it appears that a determinant cannot be irregular unless n be divisible by a square ; nor can it have r indices of irregularity unless n be divisible by a power of order r + 1. Moreover, whenever the principal genus contains but one ambiguous class, the determinant is either regular or has an uneven exponent of irregularity; if, on the contrary, the principal genus contain more than two ambiguous classes, the determinant is certainly irregular, and the index of irregularity even ; if it contain 2* ambi- guous classes, the irregularity is at least of order k, and the k exponents of irregularity are all even. A few further observations are added by Gauss. Irregularity is of much less frequent occurrence for positive than for negative determinants ; nor had Gauss found any instance of a positive determinant having an uneven index of irregularity (though it can hardly be doubted that such determinants exist). The negative determinants included in the formulae, -D = 216k + 27, =10001- + 75, = 1000 ^'+ 675, except — 27 and - 75, are irregular, and have an index of irregularity divisible by 3, In the first thousand there are five negative determinants (576, 580, 820, 884, 900) which have 2 for their exponent of irregularity, and eight (243, 307, 339, 459, 675, 755, 891, 974) which have 3 for that exponent; the numbers of determinants having these exponents of irregularity are 13 and 15 for the second thousand, 31 and 32 for the tenth. Up to 10,000 there are, possibly, no determinants having any other exponents of irregularity; but it would seem that beyond that limit the exponent of irregularity may have any value. 118. Arrangement of the other Genera. — In the preceding article we have attended to the classes of the principal genus only; to obtain a natural arrangement of all the properly primitive classes, we observe that, if the number of genera be 2'', the terms of the product (i+ro(i+r,)(i+r3)...(i+r,), in which F,. represents any genus not already included in the product of the i — 1 factors preceding l+F,-, will represent all the genera. If, then, Ai, Ai, ...,An represent any classes of the genera Fj, Fg, ..., F^ respectively. Art. 118.] BEPORT ON THE THEORY OP NUMBERS. 259 and I C I be the formula representing all the classes of the principal genus, the expression |^| = | (7| x (1 +^0 (1 +^2) ••• (1+^^) supplies a type for a simple arrangement of aU the classes of the given determinant. When every genus contains an ambiguous class, it is natural to take for A-i, A^, ..., A^, the ambiguous classes contained in the genera Fi, Fg, ...,F^ respectively. When the principal genus contains two ambiguous classes (and when, consequently, one-half of the genera contain no such classes), let Ci be the class taken as base (or, if the determinant be irregular, as first of the bases) in the arrangement of the classes of the principal genus, and let Q^ = Ci', it may then be shown that fii wUl belong to a genus containing no ambiguous class, and that the formula \K\ = \C\x{l + Q,){l+A,)...(l + A,), in which A2, ...,-4^, are ambiguous classes, represents all the classes*. In general, if the principal genus contain 2* ambiguous classes (a supposition which implies that the determinant is irregular, having k even exponents of irregu- larity, and that there are only 2**-* genera containing ambiguous classes) let it will be found that all the classes are represented by the formula \K\ = \C\x{l + a,){l + i\)...{l + Q:){l + A,^,)...{l+A;), in which A^^.^, ...,An are ambiguous classes, and Qj, Oa, ...,0^ classes belonging to genera containing no ambiguous class f. A similar arrangement of the improperly primitive classes (when such classes exist) is easUy obtained. Let 2 denote the principal class of improperly primitive forms, i.e. the class containing the form (2, 1, _^(Z>-1)); we have seen (Art. 113) that the number of properly primitive classes which, * Gauss employs a class 12, producing C, by its duplication, both when one and when two ambignons classes are contained in the principal genus. The number of classes requisite for the construction of the complete system of classes is therefore fx in either case, since C, may be replaced t The principles employed by Gauss for the arrangement of the classes of a regular determinant are extended in the text to irregular determinants. If the determinant have k uneven exponents of irregularity, the number of classes requisite for the construction of the complete system of classes is [i + k'. LI2 260 REPOBT ON THE THEORY OF NUMBERS. [Art. 119. compounded with 2, produce 2, is either one or three. When there is only one such class, the number of improperly primitive classes is equal to that of properly primitive classes ; and if | ^ | be the general formula representing the properly primitive classes, the improperly primitive classes will be repre- sented by 2x|^|. When there are three properly primitive classes, which, compounded with 2, produce 2, the principal class will be one of them, and if (/) be another of them, (p^ will be the third ; also (p and <p^ will belong to the principal genus, and will appertain to the exponent 3. When the deter- minant is regular, instead of the complete period of classes of the principal genus, 1, C, C", ..., C"~S we take the same series as far as the class (7*" ex- clusively; when the determinant is irregular, we can always choose the bases Ci, C2, ... in such a manner that the period of one of them shall contain <p and (p^, and this period we similarly reduce to its third part by stopping just before we come to (p or <p\ Employing these truncated periods, instead of the complete ones, in the general expression for the properly primitive classes, we obtain an expression, which we shall call \K'\, representing a third part of the properly primitive classes, and such that 2 x | ^' j represents all the im- properly primitive classes. 119. Tabulation of Quadratic Forms. — In Crelle's Journal, vol. Ix. p. 357, Mr. Cayley has tabulated the classes of properly and improperly primitive forms for every positive and negative determinant (except positive squares) up to 100. The classes are represented by the simplest forms contained in them * ; the generic character of each class, and, for positive determinants, the period of reduced forms (Art. 93) contained in it, are also given. The arrangement of the genera and classes is in accordance with the construction of Gauss, ex- plained in the preceding articles ; and the position of each class in the arrange- ment is indicated by placing opposite to it, in a separate column, the term to which it corresponds in the symbolic formula (such as \K\ or 2 x |^|) which forms the type of the arrangement. To the two Tables of positive and negative determinants Mr. Cayley has added a third, containing the thirteen irregular negative determinants of the first thousand. * The Bimpleet form contained in a class is that form which has the least first coefficient of all foi-ms coutained in the class, and the least second coefficient of all forms contained in the class and having the least first coefficient. If a choice presents itself between two numbers differing only in sign, the positive number is preferred. In the case of an ambiguous class of a positive determinant, the simplest ambiguous form contained in the class is taken as its representative. Art. 119.] REPORT ON THE THEORY OF NUMBERS. 261 In a letter addressed to Schumacher, and dated May 17, 1841, Gauss expresses a decided opinion of the uselessness of an extended tabulation of quadratic forms. ' If, without having seen M. Clausen's Table, I have formed a right conjecture as to its object, I shall not be able to express an opinion in favour of its being printed. If it is a canon of the classification of binary- forms for some thousand determinants, that is to say, if it is a Table of the reduced forms contained in every class, I should not attach any importance to its publication. You will see, on reference to the Disq. Arith., p. 521 (note), that in the year 1800 I had made this computation for more than four thousand determinants ' [viz., for the first three and tenth thousands, for many hundreds here and there, and for many single determinants besides, chosen for special reasons] ; ' I have since extended it to many others ; but I have never thought it was of any use to preserve these developments, and I have only kept the final result for each determinant. For example, for the determinant — 11,921, I have not preserved the whole system, which would certainly fill several pages*, but only the statement that there are 8 genera, each containing 21 classes. Thus, all that I have kept is the simple statement viii. 21, which in my own papers is expressed even more briefly. I think it quite superfluous to preserve the system itself, and much more so to print it, because (1) any one, after a little practice, can easily, without much expenditure of time, compute for himself a Table of any particular determinant, if he should happen to want it, especially when he has a means of verification in such a statement as viii. 21 ; (2) because the work has a certain charm of its own, so that it is a real pleasure to spend a quarter of an hour in doing it for one's self; and the more so, because (3) it is very seldom that there is any occasion to do it My own abbreviated Table of the number of genera and classes I have never published, principally because it does not proceed uninterruptedly.' f Probably the third of Gauss's three reasons will commend itself most to mathematicians who do not possess his extraordinary powers of computation. An abbreviated Table of the kind he describes, extending from -10,000 to +10,000, would occupy only a very limited space, and might be computed from Dirichlet's formulae for the number of classes (see Art. 104), without constructing systems of representative forms. But it would, perhaps, be desirable (nor would it * Mr. Caj'ley's Table of the first hundred negative determinants occupies about four pages of Crelle's Journal; the determinant —11,921 would occupy about one page. t Briefwechsel zwischen C. F. Gauss und H. C. Schumacher, vol. iv. p. 30. I 262 REPORT ON THE THEORY OP NUMBERS. increase the bulk of the Table to any enormous extent) to give for each deter- minant not only the number of genera, and of classes in each genus, but also the elements necessary for the construction, by composition only, of a complete system of all the classes. For this purpose it would not be necessary to specify (by means of representative forms) more than 5 or 6 classes, in the case of any determinant within the limits mentioned. rx. REPORT ON THE THEORY OF NUMBERS. Paet v. [Report of the British Association for 1863, pp. 768-786.] 120. (jrEOMETRICAL Representation of Forms of a Negative Determinant. — Before quitting the subject of binary quadratic forms, we have still to men- tion several investigations of great interest, relating chiefly to forms of a negative determinant. We shall first refer to the geometrical considerations which Gauss has employed to illustrate the nature of these forms *. Let an infinite plane area be divided by two systems of parallel lines into similar and equal parallelograms. The vertices of these parallelograms we shall call nodes ; and we observe that every system of nodes possesses the characteristic property, that if it be displaced without rotation in its own plane, so as to bring any one node into a position originally occupied by any other node, then every node wUl also occupy a position originally occupied by another node ; and the system in its second position wUl entirely coincide with the system in its original position. From this property we infer that the system of nodes admits of an infinite number of parallelisms besides the given parallelism ; i.e. that it may be regarded, in an infinite number of different ways, as dividing the plane area into similar and equal parallelograms. For let and (J be any two nodes such that no node lies on OC/ between and (J ; let P be one of those nodes which lie the nearest to the line 0(J produced indefinitely both ways, and let PP' be drawn parallel and equal to OCX; then * See Gauss's review of Seeber's ' Untersuchungen uber die Eigenschaften der ternaren quad- ratischen Formen,' in the Gottingen 'Gelehrte Anzeigen' for 1831, No. 108, or in Crelle's Journal, voL XX. p. 312; also Lejeune Dirichlet, Crelle, vol. xl. p. 209. 264 REPORT ON THE THEORY OF NUMBERS, [Art. 120. P* is a node, and OO'P'P is a parallelogram of which the vertices are nodes, and which has no other node either on its contour or in its interior ; such a parallelogram we shall call an elementary parallelogram. It is then evident firom the characteristic property of the system, that every elementary paral- lelogram supplies us with a parallelism of the system ; also we can obtain an infinite number of dissimilar elementary parallelograms ; for if Ox and Oy are the two lines of the given parallelism which intersect in 0, and if m and n are any two integers relatively prime, the intersection of the mth parallel from Ox with the nth parallel from Oy will give a node (7 such that no node can lie on 00' between and (7 ; and, again, instead of P in the pre- ceding construction, we may take any node lying on either of the two lines of the system which are the nearest to OO'. The areas, however, of all elementary parallelograms are equal. To prove this, we observe that if AOB is an elementary triangle (i. e. a triangle of which the vertices are nodes, but which has no other node either on its contour or inside it), the parallelogram OAO'B, obtained by drawing parallels to any two of its sides OA and OB through the opposite vertices B and ^, is an elementary parallelogram. For if ^0 and BO are produced to A' and B', so that bisects A A' and BR, A' and B' are nodes, and the triangle A' OB' is elementary; because if there were a node x (other than its vertices) in A'OB', we could immediately con- struct a node x (other than its vertices) in AOB. But A'OB' can be made to coincide with BO'A by a displacement without rotation ; therefore BO' A is elementary as well as AOB ; or the parallelogram AOBO' is elementary. Hence, if two elementary triangles have a common base, they are certainly equal. For if through the vertex of either triangle we draw a parallel to the base, an elementary parallelogram will be contained between that parallel and the base ; that is, the altitude of either triangle will be the distance of the base from the parallel nearest to the base; or the triangles will be equal. Again, let AOB, aOb, be any two elementary triangles, which we may suppose to have a common vertex ; if £0a is an elementary triangle, they are each of them equal to it and to one another ; if not, let x be that node contained in BOa which lies the nearest to OB, then BOx is elementary, and has the side J50 in common with AOB ; by proceeding in this manner we shall form a series of elementary triangles, of which the first is AOB, and the last aOB, each triangle having a side in common with that preceding it, whence AOB = aOh\ i.e. any two elementary parallelograms are equal. We shall next show that it is always possible to find a reduced paral- Art. 120.] REPORT ON THE THEORY OP NUMBERS. 265 lelogram, i.e. an elementary parallelogram, the sides of which are not greater than its diagonals. Let he any node ; A a node as near to O as any other ; B a node on one of the parallels nearest to OA, and as near to as any node on either of those parallels; complete the elementary parallelogram OAO'B; it will have the property required. Produce O'B to 0", making BCX' = (JB ; then AB=0(y; hut by hypothesis OA ^ OB, and OB ^ OU , OB < 00" ; i.e. the sides of OAOB are not greater than its diagonals. Again, if OAO'B is a reduced parallelogram in which OA ^ OB, it can be proved that no node lies nearer to than A, and that no node, out of the line OA, lies nearer to than B ; for, first, no node on the line 0"B0' lies nearer to than B, because by hypothesis OB S 00', OB ^ 00", and because the extremity of the perpendicular drawn from to 0"0' falls between the points of bisection of the segments 0"B and BO', or on one of those points : secondly, no node on any parallel beyond 0"B0' can lie as near to as B, for the limits of the angle AOB are evidently 60° and 120°; whence the perpendicular distance of OA from the parallel nearest to it but one is ^ OB^JS ; i. e. the distance of any node on that parallel from is > OB. If then we join any node 0, first to a node A, which lies as near to as any other node, and, secondly, to a node B, which lies as near to as any node out of the line OA, the joining lines are adjacent sides of a reduced parallelogram ; for, by what precedes, B must lie on one or other of the parallels nearest to OA. In general, a system of nodes has but one reduced parallelism, because in general there is a pair of opposite nodes AA', each of which is nearer to than any other node whatever, and a second pair of opposite nodes BB', not lying in the line AOA', each of which is nearer to than any node not lying in that line. Even if A and B are equidistant from 0, provided only that their common distance from is less than the distance of any other node from 0, the system has but one reduced parallelism. But there are two special cases in which a nodal system admits of more than one reduced parallelism. 1. If there is one pair of opposite nodes A A' nearer to than any other node, and two pairs BB', hh', equidistant from 0, not lying in the line AOA', and nearer to than any other node not in that line, the system admits of two reduced parallelisms, having one set of parallels in common, and having their common set of parallels equally inclined to the other two sets. 2. If there are three pairs of points at the minimum distance from 0, the system of nodes forms a system of equilateral triangles ; and, suppressing M m 266 KEPORT ON THE THEORY OF NUMBERS. [Art. 120. in turn each one of the three systems of parallel lines by which these triangles are formed, we obtain the three reduced parallelisms of which the system admits. That, in these two cases, the reduced parallelisms are such as we have described, and that, except in these two cases, there is but one reduced parallelism, may be inferred from the existence of a reduced parallelogram in every system, and from the properties which have been shown to belong to it. To apply these results to the theory of quadratic forms, let ax^ + 2bxy + cy'' be a form of the negative determinant — A ; let cos (io= , and with a pair of axes inclined to one another at an angle w, let us construct all the points whose coordinates are integral multiples of \/a and Vc respectively ; thus forming a nodal system. The expression ax^ + 2hxy + cy^ will then represent the square of the distance between any two nodes, the differences of whose co- ordinates are x Va and y \/6 : and the area of an elementary parallelogram will be \/A. If the transformation x = aX+l3Y, y = yX + SY, where aS — ^y= ±1, change ax^ + 2bxy + cy^ into AX^ + 2BXY+CY'' ; and if, in the same plane as before, we construct a nodal system corresponding to the latter form — the directions of rotation from the axis of X to the axis of Y, and from the axis of X to that of y, being the same — it will be found that the two systems may be made to coincide. For if we consider the point in the first system whose coordinates are xVa, y 'Jc a^B corresponding to the point in the second system whose coordinates are X\/A, Y\/C, the distance between any two points of the first system is equal to that between the corresponding points of the second system ; therefore the two systems are identical, and are either similarly situ- ated, i.e. are capable of being made to coincide by moving either of them about in their common plane, or else are symmetrically situated, i.e. are capable of being made to coincide after the plane of one of them has been turned over and applied again to the plane of the other. On comparing any two corresponding triangles in the two systems, for example the triangle obtained by giving to X and Y the values (0, 0), (1, 0), (0, 1), with the triangle obtained by giving to x and y the values (0, 0), (a, y), {^, S), it will be seen that the two systems are similarly or symmetrically situated, according as aS-fiy= -f 1, or = -1. It thus appears that a class of quadratic forms of a negative determinant may be considered to represent a nodal system, and that each form of the class Art. 120.] EEPORT ON THE THEORY OP NUMBERS. 267 corresponds to a parallelism of the system. Conversely, to each parallelism of the system a form of the class corresponds. For let Ox, Oy be lines of any parallehsm of the system, and OX, OY lines of any other parallelism, the directions of rotation from Ox to Oy and from OX to OF being the same; let also -v/a, Vc be the lengths of the sides of an elementary parallelogram in the first system, and — p=- the cosine of the angle between them ; and let Vvl \/C have the same signification with regard to the second system ; then, if ^ (x^a, ys/c), (X^A, YVC) are the coordinates of the same node P, we must have two equations of the form x = aX+^Y, y = yX+§Y, in which x and y are integral if X and Y are so, and vice versd ; hence a, /3, y, S are integral, and aS — ^y^+l; the sign of the unit being determined by the supposition we have made as to the situation of the axes with respect to one another. Also OP^ = ax^ + 2hxy + cy^ = AX"" + 2BXY+ CY' ; or the two given parallelisms are represented by two properly equivalent forms. The theorem that in every nodal system a reduced parallelism exists, has for its arithmetical expression, ' In every class a form exists in which [26] ^ [a], [26] ^ c' We thus obtain an independent proof of the theory of reduction of Art. 92 ; the geometrical signification of the special conditions in the definition of a reduced form is as follows : — K a = c> [26], the corresponding nodal system has only one reduced parallelism ; but either of the two directions in this reduced parallelism may be taken for the axis of x, consistently with the condition that the rotation from Ox to Oy should have a given direction; the condition 26^0 implies that if the angle between the axes is not right, that direction is to be assumed for the axis of x which renders the angle between Ox and Oy acute. Similarly, if a<c, but a = [2 6], the system has two reduced parallelisms, and the condition 26^0 distinguishes one of them from the other. If a = [26] = c, the system has three reduced parallelisms, which are identical and similarly placed ; the condition 2 6 > does not dis- tinguish between these, but only between the two modes in which any one of them can be taken. The number of automorphics of a class may be ascertained by causing the nodal system which represents it to revolve in its own plane round one of its nodes and examining the number of positions in which it coincides with its M m 2 268 REPOBT ON THE THEORY OF NUMBERS. [Art. 121. original position. After a revolution of 180° it wUl always do so ; but in order that it should do so in any other position, the first and second sides of its reduced parallelogram must be equal, and must include an angle of 90° or 60°, i.e. the system must be one of squares or of equilateral triangles. Hence we infer (Art. 90) that there are in general but two automorphics for a form of a negative determinant, but that for the classes containing the forms x^ + y'^ and 2x^ + 2xy + 2y'^ (or multiples of those forms) there are four and six respectively. Similarly we may investigate the conditions for the ambiguity of a class. In order that a class should be ambiguous, the nodal system representing it must be symmetrically equivalent to itself. If therefore there is but one reduced parallelogram, that parallelogram must be symmetrically equivalent to itself, i.e. it must be either a rectangle or a rhombus. When there are two reduced parallelograms, we have seen that they are sjmametrically equivalent to one another; and when there are three, they are each of them rhombs. We thus obtain the conclusion that if (a, h, c) is the reduced form of an am- biguous class, either 6 = 0, or a = c, or a = 26 (Art. 94). 121. Ai^plication of Formulae relating to the Division of the Circle to the Theoi^y of Quadratic Forms. — We have already referred to the trigonometrical solutions of the equation T' — DU^ = 1 (Art. 96, (ix).) and to the connexion existing between them, and the number of classes of quadratic forms of de- terminant D (Art. 104). If ^ is a prime of the form 3n + l or 4«-|-l, the coeflficients of the cubic, or biquadratic, equation of the periods depend on the values of the indetermi- nates in the equation 4ip = x^ + 3y'^, or p = x^ + y'^ (Art. 43). Thus in these two cases, if, for any given value of p, we calculate the equation of the periods, we obtain, by a direct though tedious process, the values of the indeterminates in certain simple quadratic decompositions of 4j) or p. But the theory of the division of the circle supplies a method equally direct and of more general application for the investigation of such decompositions in certain cases. The principles of this method were discovered by Gauss, who deduced from them the first of the three following theorems : — 'If p = in + l=x^ + y^, x= i YT n—,^odp] 33 = 1, mod 4; n2n. 11271 , , y= -\ n n — > i^od p. (Gauss, Theor. Kes. Biq. Comm. prima, Art. 23.) Art. 121.] BEPORT ON THE THEORY OF NUMBERS. 269 x= — Y^ Y^— , mod p; x = l, mod 3 ; y = 0, mod 3.' t(Jacobi, Crelle, vol. ii. p. 69 ; Stem, ib. vol. vii. p. 104, vol. ix. p. 97, vol. xviii. p. 375* ; Clausen, ib. vol. viii. p. 140.) a; = I yt — fn~ ' ^^^ P' * = 1> ™o^ *•' I ^Un. 114:71' (Jacobi, Crelle, vol. xxx. p. 168 ; Stem, ib. vol. xxxii. p. 89.) In all these foraaulae the absolute value of x is evidently < ^p ; so that x is determined without ambiguity as the minimum residue for the modulus p of the binomial coefficient. And the combination of the two congruences satisfied by x gives rise in each case to a remarkable property of the coefficient : thus, from the two congruences satisfied by a; in the first theorem, we infer Yl2n that 'if j3 is a prime of the form 4n + l, the minimum residue of ^ ^=j^ yf for the modulus j9 is of the form 4m + l.' To show, by an example, how these formulae are obtained, we shall consider the last of them in particular. Resuming the notation of Art. 30, let be a primitive root of the equation cc*'"^ — 1 = ; and let 8 = a;p _ 1 X representing a root of the equation j- = 0, and y a primitive root of the congruence x'^~^ = 1, mod2>. Then , , , Fie-"*) i?'(0-*») is an integral fimction of « only (Art. 30, ui.) ; let •v|/- (to) = a + &a) + c w^ + dco\ The fimction f//,.. n — - is iiot changed, if for 0-» we write 0-3»; there- fore>fr(ft,) = >/.((o3); 4.e. (6-c^ (l-i)« + 2a = 0, or c = 0, 'b = d, and ■v|'(<o) = a+6(l + i)ft», -^{w-^) = a-\-h{l-i)w-^; so that _p = xf' (w) X >f^ (ft)-i) = a** + 2 6^ (Art. 30, iv.). * {This reference relates io;p = 8n\ 1 =a^ + y'.} 270 REPORT ON THE THEORY OF NUMBERS. [Art. 121. ^'"' >l'(7") = a + &(7" + 7'")=-n^^.«^odi.(Art. 30, v.), whence _ HBn ~ ^II«..Il4n To show that a = — 1, mod 4, we observe that by the definition of the fiinc- tion ■^, \|/-(w) = Sft)"''i~*''», yt ^^^ 2/2 representing any two numbers of the series 1, 2, ...,p — 2, which satisfy the congruence y'l + 'y^'asl, modp. Hence a = S( — l)''a+''i, where «7i is one of the numbers 1, 2, ..., 2n — 1, and i/i, y^ satisfy the congruence 7*''i + 'y''2 = l, mod^. Let a- be any one of the numbers 1, 2, ...,« — 1, and let A, B be the values of y^i corresponding to the values w — 0-, » + (T of >?i ; then therefore ^ x 5 is a quadratic residue of p, and the values of 3/3 corresponding to the values n — a-, n + a- of i/i are either both even or else both uneven ; also, if *ii = n, •y^2-=.2, raoAp, and y^ is even, because (~\ = 1. Let k be the number of values of >7i, included in the series 1, 2, ...,n — 1, for which 2/2 + "71 is uneven ; then a = 2(-l)«'2+'?i = 2(n-l)-4^ + (-l)»; i.e. a= -1, mod 4. We might also determine x in the equation p = 03^ + 21/^ by the congruence or by the congruence , ^ = 2^»^x "^n ^^ •^ II ft, lift -^ These determinations, which have been given by M. Stem, may either be obtained directly by considering the functions i?-(e-») j?^(e-3'») [F(e-")]g or may be deduced from the formula of Jacobi. The formulae for the deter- mination of X in the first two theorems also admit of various modifications. It will be observed that, in the first, y is determined by a congruence as well as x. This determination is obtained by a comparison of the two congruence 1+^=0, mod p, l + (n2w)2 = 0, mod^, Art. 121.] EEPOKT ON THE THEORY OP NUMBERS. 271 (the latter arising from Sir J. Wilson's theorem) ; with regard to it Gauss observes, 'quum insuper noverimus quo signo affecta prodeat radix quadrati imparis, eo scUicet ut semper fiat formae 4m +1, attentione perdigniun est, quod simile criterium generale respectu radicis quadrati paris hactenus inveniri non potuerit. Quale si quis inveniat et nobiscum communicet magnam de nobis gratiam feret.' These congruential determinations possess great interest, not only because direct methods of solution present themselves very rarely in the theory of numbers, but also on account of the singular connexion which they establish between certain binomial coefficients and certain quadratic decompositions of primes. Nor is it less remarkable that the properties of the resolvent function of Lagrange form the intermediate links in this connexion ; although it is proper to observe that Gauss has exhibited his demonstration of the theorem relating to the equation p = x^ + y^ in. a form in which its connexion with the theory of the division of the circle is disguised. Results of a more general character have been obtained by Jacobi and Cauchy. Cauchy has treated of the subject with great fulness of detail in his Memoir on the Theory of Numbers, in the 17th volume of the Memoirs of the Academy of Sciences (pp. 249-768) ; while Jacobi has barely indicated his method in his note on the division of the circle (Crelle, vol. xxx. p. 166) ; nevertheless, as in some respects it seems preferable to that employed by Cauchy, we shall endeavour to adhere to it in what follows. Retaining the other notations which we have employed in this article, let Jr^Q-^Jn)) ' ^^^i^.^e) or ^^(m,w), when there is no occasion to consider 6 explicitly; we observe that >f.(m, n) = >!'(«, m); >|. (0, to) = xf. (m, 0) = >|. (0, 0) = - 1 ; also y^r (m, n^ = -<^{m, n), if m'=m, mod p — 1, n=n,Taodp — l; ■^(m,n) = {-l)"**^p = {-iy*^p, if m + n = 0, modp-1, but m and n are not =0, mod j)-l. Let Wj, m,', ...,^1^"^ be any set of cr + l numbers, each of which satisfies the conditions < w^ <p — 1 ; let mi + mi'+...+mi<''' = «i(j)-l) + Si, where < Sj < j) — 1 ; and put F{e-^i) F{e-'"i) ... 2^(0-"'/''') = x(^) F{6-'i). Writing, for brevity. 272 REPORT ON THE THEORY OP NUMBERS. [Art. 121. and determining fjil, fi^', ft", ... so as to satisfy the conditions (i^ni<p — l, we find X{Q) = ^ (Ml, mO ^ (mx', mO ...>\r (m,«'-", w,")). In this expression if /«/'''+ mi^*+i'>p — 1, we write for ^{iJ-^^,'m^**''^) its equi- valent ^-r>|'(_p — 1— /Ui^'"), p — 1 — mi<*+'^) ; and if iM^^ + m^*'^'^'>=p — l, we write for y\f{iJi^^,im^**'^^) its equivalent (-iy+'"i'"*'''j9. It is evident that the con- dition M''^-f-m<*'+'^>p — 1 wUl be satisfied Uy times precisely; so that x(^) assumes the form j?"i .^ .J^ , ^i{&) and ■^i(0) denoting products of factors of the form ■^ {h, h'), in each of which h + h' <p — l. It will now be found that £lW = ( _ !).-», ^fi ^^ , mod p. For (1), if A(i'''^-|-mi^''+^'<p — 1, we have >|.(m,« W^^', 7)= - n^"nm%^) > ^°^^' (2), if /*!<*■> +mi^'+i>>^- 1, we have n(j9-i-Mi'''^).n(j?-i-mi»+^>) _ n^u^^+i) n (2p - 2 - Mx'*'^ - mi«+ D) - n^i*'^ . nmx('+ ')' ^' since, by Sir J. Wilson's theorem, n(i>-l-i)=- jj / ,modj), if ;<^-l; (3), if /^jM + m/' + 1^ = J) - 1, we have ,. . TTh (»' + i' (_lV + '»i <* + !'= -^ mod» because n/UiWnmi«+i' = (-l)i+'».""^'\ mod_p, by Sir J.Wilson's theorem, while Il/Ui''^-i' = 1, since /x/* + i' = 0; whence, mul- tiplying and writing 5^ for n^^"^, we obtain the congruence written above. Let r represent any term of a system of residues prime to p - 1 ; let the numbers mr,mr', ..., m/"^ be determined by the congruences m}''^=.m^*^r, mod(p — 1), combmed with the condition < m,^^ <p-l; and let m^ 4- m/ + . . . 4- m/"^ = n,. (^ - 1) -I- 5,. , where again < s,. < ^ — 1 : we have for every value of r an equation of the form and a congruence of the form xW-P"'x||>, ?ri2^=f_lV-nr ^L^ modw Art. 121.] REPORT ON THE THEORY OF NUMBERS. 273 Let x{^) = ^o + AO+...+A,,d\ k+1 denoting the number of terms in a system of residues prime to ^ — 1 ; let riy be the least of the numbers n^, ..., % + i, and y the exponent of the highest power of p dividing Ag, A^, ..., A^: then shall j = n^. For, first, if y>?i,, from the equation "^^ (0) ^\ = ^^ (0), in which the coeflScients of the powers of 6 are integral numbers, we infer the congruence %(y)^ = ^y(y),modp; Tc (y") but n — ^' ™°^ P ' therefore, ^^ (y) = 0, mod p, which is impossible. Secondly, if y < «„, writing A/ for Ai -f p', and observing that "^r (?) is prime to p, for every value of r, we find AJ + A{y' + A.;y^'+...+Ak''/'-=0, modp\-\ for every value of r : but the determinant of this system is prime to p, therefore Ao=0, Ai=0, ..., Ajc' = 0, mod p\~', which is contrary to the hypothesis that j KUy, and that jp-* is the highest power ot p dividing A^, A^, ..., A^. The application of these results leads to the following general theorems ; in the enunciations of which p) is an uneven prime, and A a number not divisible by any square. 'If A = 4m + 3, p = ^n + \, and if we represent by a and h numbers less than A and prime to A, respectively satisfying the equations (1-) = ^' ( A ) = - 1> we tave S6-2a iA^p ^ =03^ + Ai/^, ' If 2) = 4 An + 1, A being of any other linear form than 4m + 3, and if we represent by a and 6 numbers less than 4 A and prime to 4 A, respectively satisfying the equations ( ) = + 1, (~7~} = ~ 1' w® h&\Q 26-20 (p ^ =a;2 + A2/2, (B), ^ 1 In these formulae the signs of summation extend to every value of a and h N n 274 REPORT ON THE THEORY OF NUMBERS. [Art. 121. respectively; and in the expression n„[nan] the exterior sign of multiplica- tion Ha extends to every value of a, whUe the interior sign is the factorial symbol, so that Uan = l .2.3 ...an. The number 3 is excluded from the first formula ; the numbers 1 and 2 from the second. It will suffice to show how the first of these two theorems is to be de- monstrated. For this purpose we consider the product UF{6~'*'^); taking an, a'n, a"n, ... for mj, m^', ... we find x{&)= -11^(0-""); because (as may easily be proved) 2a = 0, mod A, whence 2a» = 0, modj> — 1. We shall now show that x(^) is of the form A'LQ"'* + B'LQ^^. Actually multiplying the expressions F{d-'"*), F{6-''^), ..., the coefiicient, in the product, of any term such as a^fl™" is equal to the number N of the solutions of the simultaneous congruences y^ + y^ + y^'+ .•• = h mod p, ay + a'y + a"y" + . • • = - «», mod A. If r is a number prime to A, and satisfying the equation (-r-) = ■\-\, N will not be changed, if we write rm, ra, ra', ... (or rather the least positive residues of those numbers, mod A) for m, a, a', .... Hence, in x (^) all powers of 9 whose exponents are of the form an have the same coefficient A', and all powers of 6 whose exponents are of the form bn have the same coefficient B'. Again, con- sider a power of 6 of which the exponent is of the form aSn; § representing a given divisor of A (other than 1 or A), and a representing any number less than A . A -T- , and prime to -»- ; all such powers of 6 will have the same coefficient. For we can always find a number r prime to A, satisfying the equation (-ir) = ^, and yet congruous, for the modulus -j- , to any given number prime to -j- ; whence it follows that the number N will remain the same for all values of m included in the formula aS. But a sum of the form 2„0°'" is equal to -1-1 or — 1, according as the number of primes dividing -y is even or uneven, be- A cause it is the sum of the primitive roots of the equation a;' = 1. Thus, the function x {&) assumes the form ^'2 fi"" -|- ^ 2 0*» -f- C", whence, attending to the equation 20'"'-|-20*» = ( — 1)\ in which X is the number of primes dividing A, we find, as has been said, -nir(0-«») = x(0) = ^26''» + j520K Kwe write 0~^ for in this equation, it becomes -ni^(0'"')=x(0-O=^2:0*»-i-52e«», Art. 121.] REPORT ON THE THEORY OP NUMBERS. 275 Multiplying the two equations together, and observing that P{Q-an^ i?'(0«») = ( - 1)«»^ =p, because n is even, we obtain 4_p^^o(^) = [( - 1)^(^ +B) + {A-B) (20«» - 20^")] y.[{-\Y\A + B) + \A-B) (20''» - 20™)], ;or, since (20«» - 20*»)2 = -A*, 4_p^^o(-^) = (^A+BY + ^{A- BY, ■^^(A) representing the number of numbers less than A and prime to A. We have next to determine the highest power of p dividing A+B and A—B, or, which is the same thing, A and B. By the principles indicated above, we have 2a ^ /a\ A^e^^+Bi:6^=p^ ^^, ^1 (y) ^ 20^ + 520O" = p"^ ^^^1 . ' Writing in these equations y for 0, and observing that the determinant (Zy'"'Y - 2 {y^^'Y t, as well as the four numbers *i(7), *-i(7), %iy). ^-1(7), is prime to p, we infer that the exponent of the highest power of p dividing A and B is the less of the two numbers -r- , —7- • Of these the fonner is A A the less t ; if therefore we write x and 1/ for {-1Y{A + B)p~~^, and {A-B)p~^ * See Art. 96, (is.) of this Report, or the note on Art. 104. t Since 2fl''" + 29'« = (— 1)*, we have Syn + Sy*" = (-1)*, modp; and since (Sfl""— 20*")'' = — A, we have (2y«"-2y'"')== — A, mod p. Thns the two factors of the determinant are each of them prime to j). The principle that any rational equation containing only powers of d and integral numbers may be changed into a congruence for the modulus p, if y be written in it for 0, has already been employed in this Article. Its truth is evident, if we observe that the irreducible equation satisfied by d, if considered as a congruence for the modulus p, is satisfied by y. This principle is of more general application than a similar one which has been already employed in Art. 5 1 of this Report ; but its proof supposes the irreducibility of the equation of the primitive roots, which is not necessary to the proof of the principle of Art. 51. X 26— 2a is certainly positive, because -r — is equal to the number of improperly primitive dasses of the negative determinant — A. Or (as it is desirable to avoid making use of this result N n 2 276 REPORT ON THE THEORY OP NUMBERS. [Art. 121. respectively, our equation becomes Also, since ^i (y) = _ / _ i xx 1 we have _^ _?f ?^ i ^jj ASyi^ + ^p ^ 27«» = 0, modjp, here) 26— 2o is positive because (26— 2a) is the sum of the series 2 ^— j -, the summation extending to every value of w prime to A, and the terms being taken in their natural order. This series is positive, because the series 2 (-jr) -j+i , of which it is the limit, when p is diminished without limit, is certainly positive, being the reciprocal of the product in which the sign of multiplication extends to every prime q not dividing A, and in which every °° / w \ 1 factor is positive. The series 2 ( —- ) - is one of those summed by Dirichlet in the memoir 'Re- ^ 1 VA/ n cherches sur di verses applications &c.' (Crelle, vol. xxi. p. 141 et seq.) : for the case in which A is a prime, he had already summed it in the memoir on the Arithmetical Progression (Memoirs of the Academy of Berlin for 1837, p. 55). Cauchy (M^moires de I'Acad^mie des Sciences, vol. xvii. p. 673 et seq.) inverts Dirichlet's process, and transforms sums of the form 2/ (o)— 2/(6) into infinite series. The transformation is effected by substituting for /(a;), in the expression Y(|-)/W = S/(a)-S/(6), the equivalent infinite series whence, observing that we obtain i VA (/(a)-/(6)) = '"^"i'^) f^ sin ^^/(«) ds ; m=l ^ ^ Jo ^ a formula from which Dirichlet's result is immediately deducible, by putting /(.'») = x, and performing the integrations. It is a remarkable fact that the inequality 26>2a has never been proved by elementary considerations, or witliout the use of infinite series (see the Memoir on the Arithmetical Progression, p. 57). If A is a prime, 25 — 2a is certainly not zero, for 26 + 2a is uneven (because A is of the form 4m + 3); but even this remark cannot be extended to the case in which A is composite. Art. 121.] REPOET ON THE THEORY OF NUMBERS, 277 whence by addition Xb mod p. n„[naw]' If A is a prime, x also satisfies the congruence ^x = l, mod A; for the sum of the coefiicients in any function ■v|/(m, n) is =— 1, modp — 1, and therefore mod A ; whence the sum of coefficients in x (^)) which is a product of an even number of such functions, is = 1, mod A ; because the reduction of X{&) to the form A'ZQ"" + Blid^ is effected only by means of the equations 0«A_1=O, 0»-l ; whereof the former does not alter the sum of the coefficients at all, and the [latter alters them only by a multiple of A. Consequently i(A-l)(^+5) = l, mod A, So |or, smce ^^=1, modA, and (-1)^=-1, lx = l, mod A. It wUl be observed that if A is of the form 8m + 7, whether A is a prime or not, X and y are necessarily even in the equation (A) ; whence, dividing by 4, we may put the equation in the form 2i-Sa ^ =x^ + Ay^ P Z6 ^- ( l)^2n,[nan] , mod^. Ex. Let A = 7, j) = 7n + l; the values of a are 1, 2, 4 ; of 6, 3, 5, 6 : hence 1 , nSri p = x^-[-ly\ x=-\. = 1. -,mod^; also re = 1, mod 7. ■JlnJl2n.Tl4:n~^lln.n2n' {Jacobi, CreUe, vol. ii. p. 69.) Whenever the exponent of p is 1, the formulae (A) and (B) completely de- termine the value of x ; when the exponent of p is 2, we can only be sure that the absolute value of x is less than p, so that x is not completely deter- mined, but is either the least positive or the least negative residue of Ihe binomial coefficient ; though in this case if A be a prime of the form 4 » -I- 3, the ambiguity may be removed by the congruence \x = 'i, mod A. But when the exponent of p is > 2, x is never completely determined by the congruence for the modulus p. It is very remarkable that the exponent of p in the formula (A) is pre- cisely the number of improperly primitive classes of determinant —A, and in 278 REPOBT ON THE THEORY OF NUMBEBS. [Art. 121. the formula (B) is precisely the number of properly primitive classes of deter- minant — A*. ♦ See Art. 104 of this Report. When A is of the form 4»+3, the two expressions given by Dirichlet for the number of properly primitive classes of determinant — A are where A and B represent the nombers of residues inferior to ^A, and satisfying the conditions (-r-) = + 1 and (-T-) = — 1 respectively. Hence — - — is the number of improperly primitive classes ; because that number is equal to or is one-third of the number of properly primitive classes, according as A = 7, or = 3, mod 8 (see Art. 103 or 113). n There is no difficulty in showing that Dirichlet's two expressions are identical. If ( — ) = 1, the congruence 25'= 6, mod A, is always resoluble; and if 6 receive in succession all positive values less than A which satisfy the condition f — ) = — 1,6' will obtain the same values in a different order. Hence S — = 2 — 2 — = S — AAA A But if 5'< iA, 26'-5 = 0; if5'>iA, 26'- 6= A, i.e. 2^ = A, for there are A values of 5 greater than J A. Similarly 2 — = 5, so that = A—B. In -r-) = — 1, by considering the congruences 25'+ 5 = 0, mod A, 2a'+ a = 0, mod A, that S?^ = 2'^ = 2A + B and "^ ==2'^^ 2B + A; A A A A 1 « 26— 2a . „ whence 3 r = A—B. A Also the expression given in Art. 104 coincides with A—B. For that expression may be written in the form a' and 6' representing numbers less than J A. IT \F(0-'")'\^ If we consider, as Cauchy has done, the product r, p,/a~2an\ > ^^^ exponent of p in the formula 11 /'(e "") U{F(0~<"')V (A), will be A—B. That product is evidently equal to nF{d-<"'), or to Jr r, V^-t-V ' according / 2 N 11 / [fl ) as Ijr) = + lor= — 1; a result which is in accordance with the equation In the formula (B), the exponent of ;j, obtained by the consideration of the same product, is A' and J5' denoting respectively the numbers of residues of the classes a and 5 respectively, which are inferior to 2 A. Art. 121.] REPORT ON THE THEORY OP NUMBERS. 279 Before Dirichlet's discovery of the formulae expressing the number of classes of quadratic forms of a given determinant, Jacobi, having succeeded in determining the exponent of ^ in the formula (A), for the case in which A is a prime number, was led with singular sagacity to conjecture that -r — - must represent the number of improperly primitive classes of deter- minant — A*. If A is the number of classes in the principal genus of impro- perly primitive forms of determinant —A, it follows from the theory of com- position of quadratic forms that 2j9* can always be represented primitively by the principal form in that genus, i.e. by the form (2, 1, J (A 4-1)), and that the exponent of the lowest power of p which is capable of such representa- tion is either A or a submultiple of h. Again, the equation 26-Sa if we write in it 2 X -I- F for a;, and Y for y, becomes 2p A =(2, l,i(A-H))(Z,F)^ the values of X and Y being integral. Assuming, then, that there exist primes of the linear form nA + 1, the doubles of which are capable of repre- sentation by a class appertaining to the exponent h (an assumption which impUes that —A is not an irregular determinant, at least in respect of its improperly primitive classes), we see that in the case in which A is a prime of the form 4« + 3, and in which therefore there is but one genus of impro- perly primitive forms, -r must be equal either to the number of im- properly primitive classes, or to a multiple of that number ; and as Jacobi found, upon a sufficient induction, that h was always equal to -r , he did not scruple to enunciate the theorem as true. We know, however, from an account which Dirichlet has given of a communication made to him by Jacobi, that Jacobi never obtained a demonstration of the theorem ; and, indeed, it * Crelle, vol. be. p. 189. Jacobi counts the classes of the prime determinant —A on the prin- ciple of Legendre, not distinguishing opposite classes from one another. If n is the number of improperly primitive classes bo counted, we have 7j=2w— 1, because there is but one improperly primitive ambiguous class. When A is of the form 8 m +7, Jacobi enunciates the theorem with reference to the number of properly primitive classes, which in this cdse is equal to the number of improperly primitive classes. 280 REPORT ON THE THEORY OP NUMBERS. [Art. 122. would seem probable, as has been observed by Dirichlet, that its demonstration requires other principles (Crelle, vol. lii. p. 206). It is hardly necessary to add that when there is more than one genus of forms of determinant — A, {. e. in every case except when A is a prime of the form 4w + 3, the exponent of p in the formulae (A) and (B) is always a multiple of the least exponent for which those formulae can be satisfied. 122, Extension of the preceding Theorem by Eisenstein. — In the theory of which an account has been given in the last article, the prime number p \a B throughout supposed of the linear form wA + 1 or 4nA + l; thus in the equa- tions p = x^ + 7y^, p = x^ + 8y^, we have supposed p to be of the forms 7n + l and Sn + 1 respectively. But we know that some power of every prime of I which —A is a quadratic residue is capable of representation by the form | x^ + Ay^; and, in particular, that primes of the form 8n + 3 are capable of ' representation by x^ + 2y^, and primes of either of the forms 7n + 2 or 771 + 4 by x' + 7y^. M. Stem found by induction that the value of a; in the equation ^ = 8/1 + 3 = 03^ + 22/2 satisfies the congruences ^^-^nf.nst+r"^"^^' ^=(-i)"> «^od4*; and Eisenstein succeeded in demonstrating this theorem, as well as the two following t : — 'I£ p = 7n + 2 = x^ + 7y\ ^=*lTO^'"^°^^' x = 3,mod7;' 'Ti p = 7n + 4: = a^ + 7y\ '" = Mln.lI2n + r "^"^^> 0^ = 2, mod 7. These demonstrations are obtained by expressing the prime number p as the product of two complex factors, composed of 8th or 7th roots of unity. But the decomposition of p is no longer supplied by the formula of Art. 30 ; nor are the complex factors included in the definition of the functions ^^j which have been considered in Art. 30 and in the last Article. If p = Sn + S is a real prime, p is also a prime in the theory of complex * Crelle, vol. xxxii. p. 89. We enunciate the latter part of the theorem in the form in which it has been given by Eisensteiil. t Crelle, vol. xxxvii. p. 97. Art. 122.] EEPORT ON THE THEORY OF NUMBERS. 281 numbers of the form a + bi; let 7 be a primitive root of ^ in that theory, and let yo = 1 +iz, mod p, z representing one of the real integers, 0, 1, ...,j)-l. jAlso let ■vf' (w) = Zw", w denoting a primitive 8th root of unity, and the sum- mation extending to every value of y. Eisenstein establishes the equations [ -whence -^ (w) is of the form A->rB{\-\-%) w, and 'To find the residue of A, mod p, let e = |(^^ - 1) = 3 n + 1 + W2? ; [and write successively 7* and 7"* for w in the function \p- (w). We find >i^ (7") = 27'"' = 2(1 + Of = 2 (1 + tz)3» + 1 (1 - izf, mod p, [because in general (a + hiy = (a — hi), mod p. In this expression no power [of z has an exponent divisible by ^ — 1 ; but \ 2 z* = 0, modp, 15 = unless Q is different from zero, and is a multiple of p — 1 ; therefore ■^f (7*) = 0, mod p. tAgain, because 5e = 7n+ 2 + (5w + 1)|>, ■^ (7''«) = 2 (1 + z^y " + ' (1 - zt)5» + S mod j) ; this expression the coefficient Cof z^"^ is vr iv^"' n77i+2.n5n+i "'^ ' n/x.n7n+2-M.n/.n5»+i-M'' [where /x + /i'=j9 — 1, and the summation extends from M = 3/i + l to /x = 7w + 2. [Writing 3w + 1 + »< for m, 5 to + 1 — v for m', and observing that Hm . nM'= ( - 1)1 +'^, mod _p, we find v=4« + i n7ft + 2.n5n + l _ n7ft + 2.n5n + l ■'^^n + i n4TO + l ^= ,Io n4ft+i-i'.nv - n4«+i ^ Jl^ nv.n4w+i-./ _ n7TO+2.n5TO+i _ n4n+i = — n4n+i ""^ -n».n3TO+i'"'°'*^' observing that 2*" + ^ = — 1, mod p, and transforming each of the three factorials by Sir J. Wilson's theorem. Hence, finally, , T74« 4-1 00 282 BEPOET ON THE THEORY OF NUMBERS. [Art. 122. in accordance with the enunciation of M, Stem. The congruence ^=(-1)", mod 4, is inferred by Eisenstein from the values of but we may omit these determinations here. Ifjp = 7n + 2, or 7n + 4, Eisenstein considers the complex numbers formed with the roots of the equation i?^ — 21>; — 7 = 0. If w is an imaginary seventh root of unity, and ;7j. = 3((o* + <i)~*) + l, the roots of this equation are 171, rj^, 173; and every complex number formed with them is of the type a + b>ii + cti2, ci, b, c denoting real integral numbers. Let 7 be a primitive root of p in this complex theory (jp is a prime of the theory, because the congruence ^^ - 21 ij - 7 = 0, mod p, is irresoluble : see Art. 44 of this Report), and let 7«' = l+«ii;i + Z2'72, mod p, Zi and «2 each representing any term of the series 0, 1,2, ..., p — 1. Tlie function y^f (w) = Hco" (the summation extending to all the p^ values of y) is shown by Eisenstein to satisfy the equations x|.(a,) = x|.(co^) = x^(«^), >Ka,'') = xf.(o,«) = x|.(a,«), ^ {w) X ^ {<o-^) =p ; whence -^ (w) is of the form a + h {w + td' + w*) + c {w^ + w^ + w"), and p = A^ + lR; i^ A = a — \{b-\-c), B = \(b — c). The equation 2>^ = a-t-3& + 3c, considered as a congruence, mod 7, becomes A =p^, mod 7 ; i.e. A= 4, or = 2, mod 7, according as p is of the form 7m + 2 or 7m + 4. To obtain the congruence, mod p, which is satisfied by ^, we consider the congruence 2A = ^ (y') + y^ (y^'), mod p ; in which e = ^(p^ — l) = a + ^p + yp^, a, /3, y representing positive integers less than p, of which the sum will be found to be p - 1. Now -^ (7') = 27*'' p—ip—i = 22 (1t-%i7i + Z2«72)«x(1+2iI7„ + Z2»;2„)^x (l + 2i»7„« + 22i72„»)')', mod «o; because in general ,- „, s-, „, . Hence y\f (7*) = 0, mod p, because a + j8 + 7=^ — 1, and because p— ip— 1 2 2 z^iz^i=0, mod p, unless ^1 and 6^ axe both different from zero, and both divisible by p — 1. Art. 123.] EEPORT ON THE THEORY OF NUMBERS, 283 Again, ]£Se = a.' + ^p + yp', we find a+^ + y=''(p-l); and omitting terms in which the sum of the indices of Zj and z^ is inferior to 2 (p — 1), p-i p-i y^r (t-S*) =2 2 (Zi >7i + 22 ./a)" (Zi rip + Z^ ri^^& (Zj ;? 2 + z^ n^J)i^ mod «. Substituting for Zj j;^^ + z^ m^^ its value — Zi j/j - Zg »;2 — Zj f/j, — Zg ijzp, we obtain ^ (7='') = Ha'. n/T. Hy'.^i ^2 (zi ^1 + Z2 ^2)^-^ (2i »;, + 22 ';2p)^-S mod p ; because every such sum as ^2 ^2 (zi.?i + Z2.;2)''-i + «(Zi>;^ + Z2^J*-»-'', in which Q is one of the numbers 1, 2, 3, ...,^ — 1, taken positively or negatively, is certainly = 0, mod p, as may be seen by substituting {z^ r)^ + z^ tj^p) for (Zj vi + Z2 172)^. Lastly, the coefficient C of (zj Za)**"^ in the expression p-ip-i 2 2 (ZiVi + Z^ri2V~H^1p + ^2l2py-^ \ 00 is evidently "2 ^Ki(r„,,,y(v,v,y-'->', .ffj, representing the coefficient of cc'' in the expansion of (1 + xy~^. Hence C = {rii ri^p - izipY''^ = 1, mod p, because »7i»?2p — '?2'?j)= ±21 ; so that finally A = iUa'.Tl^.Uy\J: ^2 (ziZ2)''-i=ina'.n^.n7', modj); an expression which, on substituting for a, /3', y' their values in the two cases p = 7n + 2, p = 7n + i, will be found to coincide with the formulae given by Eisenstein. There can be no doubt that the principles of this method are capable of many other applications ; but nothing has as yet been added to these researches of Eisenstein. 123. Applications of Continued Fractions to the Theory of Quadratic Forms. Representations of a number by quadratic forms are in certain cases deducible from the development of its square root in a continued fraction. If A is any number not a square, -^-^c the (n 4- l)th complete quotient in the develop- ment of -yA, and -^ the convergent fraction immediately preceding that complete 2 284 REPORT ON THE THEORY OF NUMBERS. [Art. 123. quotient, so that^* — .4g'* = (-l)''Z>„, then the form {ql, —p„, A), of which the determinant is ( — l)"i)„, is either properly or improperly primitive, and belongs in either case to the principal genus of its order. If we investigate the trans- formation by which this form is reduced to the simplest form in its class, we shall obtain, by an operation exempt from all tentative processes, a repre- sentation of A by that simplest form. The following proposition, however, supplies a method by which, when q^ is uneven, and (ql, —p„, A) belongs to the principal class of properly primitive forms, or when q„ is even, and {^ql, —p„,2A) belongs to the principal class of improperly primitive forms, we can frequently infer from the development of VA. itself the solution of the equations ' If (a, h, c), (a, h', c) are two primitive forms of the determinants D and D', whose joint invariant clc —2hh' + ca' is zero, and if m and m' are the greatest common divisors of a, 26, c; a, 2h', c ; then m^D' and m'^D are capable of primitive representation by the duplicates of (a, 6, c) and (a, h', c) respectively.' Thus if {a, h', c) is properly primitive and ambiguous, D can be repre- sented primitively by {1,0,— D'); if {a,h',c') is improperly primitive and ambiguous, 2D can be represented by (2, 1, ^(1— Z>')). For {a, b, c) and {a, h', c) let us take (1, 0, —A) and (q„, -p„, q„A), whose joint invariant is zero, and of which the first is properly primitive ; while the second is properly or improperly primitive according ss q„ is uneven or even, and has for its duplicate in the former case (ql, —p„, A), in the latter 2x{^ql,—p,^, 2 A): so that it is ambiguous in both cases alike. Further, let us represent by q>, q,-i (e,, —Sg, e,_i) the form into which (q„, — jp„, ?„-4) is transformed by we infer, from the property of the invariants, the equations (-l)» + ^I>„ = e.e,_j-^, e,_,D,-2S,J,-e,D,_, = 0. Let us first suppose that n is uneven, so that ( — 1)"Z)„ is a negative deter- minant which we shall call — A ; since qn (?„ x" - 2p„ xy + q„ Ai/) = (q„ x -p„ yf -t- £^if, it is evident that when q„x^ — 2p„xy + q^Ay^ attains its minimum value. X is a convergent to — ; not, we may add, the last convergent, if the last integral quotient in the development of — is unity. If therefore (g'„, —pn, qn^) 3n Art. 123.] REPORT ON THE THEORY OF NUMBERS. 285 is properly primitive and of the principal class, we shall have, for some value of s, e, = 1 ; whence A-i = - 2 ^, J;+ e,_i A, and A=JJ + D, A-i = {J. - S, D,f + AD?. If {qn,-pn,qn^) IS improperly primitive, and of the principal class of its order, we shall have for some value of s, 6^ = 2, Dg_i= —^,Js + \e^-iD„ 2^ = 2(J,-l(^, + l)Ar + 2(j;-l(^3 + l)A)A + i(^ + l)A^ We may therefore enunciate the theorem : ' If — is an inferior convergent to -yA, and A = 2M-K; when {q„,-p„,qnA) is of the principal class of forms of determinant — A, ^ is of the form X^ + A Y^, and Y is the denomi- nator of a complete quotient in the development of \/^ ; when (g'„, — ^„, g'„^) is of the principal class of improperly primitive forms of determinant —A, A is of the form 2X* + 2XF+^(A + 1)F\ and Y is the denominator of a com- plete quotient in the development of »yA.' When {qn,—pn,qnA) is ambiguous and properly primitive, but of some other class than the principal class, we must distinguish between two cases, that in which the reduced form equivalent to {q„, —p„, qn^) is itself an am- biguous form, and that in which it is of the type (a, h, a), In the former case we shall arrive at a form (e,, —S„ e,_i), in which e,, being the least number which can be represented by {q^, —p„, g'„ A), is a divisor of 2^,, and consequently of D, and 2 A ; and we shall find A = (J,-S^^) -fA^. In the latter case we shall, in the series of forms (e,, — Sg, e,_i), arrive at a sequence of one or other of the three types : (1) {2[a-h],-[a-h], a ), (a, (a - 6), 2 [a - &]) ; (2) ( a ,-[a-&], 2[a-6]), (a, 6, a ) (3) ( a , —h, a ), (o^ - & , « ) i.e. we shall arrive at a form in which e, is the least number but one, which can be represented by {qn,—Pn> qn-^)> ^^^d is a divisor of D, and 2 A ; we shall then find D^ (1) A = {J, -^D,Y + A^; (2) A = {J,,, + \D;)' + A^; (3) A = {J, +iD,y + A^. 286 REPORT ON THE THEORY OF NUMBERS. [aH. 123. Similar results may be enunciated for the case in which (g'n> "JPni Q'n-^) is improperly primitive and ambiguous, but not of the principal class. In applying the preceding formulae to particular cases, the following theorem of Goepel's is very useful. Since ^> = 9nP^,-^PnP.q. + Aq„ql -S, = q„p,p,_i-p„{p.q._i+p,_iq,) + Aq„q,q,_^, we find, if fi, is the integral quotient immediately succeeding ~, that ^«+i = ^< "/*»««• Hence ^i, 4> ••• form a continually decreasing series. But Si=pa is positive, and S„= — Ag'„_i is negative; there exists, therefore, a pair of consecutive terms S, and 5,+i, of which the former is positive, or zero, and the second negative ; Goepel shows that — ^g ^,^.i < A. For we find q, f,-i + q,-i Ss = {- 1)' {qnPs-i -p» qs-i), q. s> + q.-i f« = ( - 1)"" ' iq»Ps -Pn q.) ; ie q,^>.i+q.-iS. _ 1 ■ . whence q,e.-i+q,-iS, >M«(9',^, + 9',_ie,); or multiplying by e„ Aq, > -S, 8,^.iq, - e, 5,+i q,_i ; that is, A > —S,Sg^.i, because Sg^i is negative. Thus if A = 1, we have necessarily ^. = 0, e. = e,_i = l, D,_, = D„ A=J^ + Dl If A = 2, we have either (1) S, = 0, e,.i= 2, e. = l, A_i = 2A, A^Jf + 2D]; or (2) ^, = 0, e,_,= 1, e. = 2, D, = 2D,.„ A = J.' + 2D^_,; or (3) ^, = 1, ^«+i=-l, M, = 2, e,= l, e,_i = e,^i = 3; A-i = 3A-2J., A = {J,-D,y + 2D^ = {J,,,-D.y + 2Df. If A = 3, we have either (1) ^, = 0. e._,= 3, e,= l, A-i = 3A, A=J^ + 3D!; or (2) ^, = 0, e,.i= 1, e, = 3, D. = 3D,_„ A=J,' + 3D^_,; or (3) S. = l, ^,+i=-2, M, = 3, e, = l, e,_, = 4, D,_, = iD,-2j„ A = {J,-D,y + SD^; or (4) ^, = 2, 5, + i=-l, M, = 3, e, = l, e, + i = 4, Z),+i = 4D,-2J',^i, ^ = (J.,j-Ar + 3Z>?; or (5) ^,= 1, ^,^.1=-!, n,= 2, e, = l, €,_i = e,^.i = 4, i),_i = 4D,-2 J"„ ^ = (J,-Z),y + 3A^; or (6) 5,= 1, 5, + i=-l, fi, = l, e, = e,-i = e, + i = 2; D,_j = D,-J, = J,^.^, Art. 123.] BEPOBT ON THE THEOKY OP NUMBERS. 287 the last case occurring always and only when q^ is even. If A = 7, and if we suppose q„ even, so that (qn,-Pn, ^n^) is improperly primitive, we shall cer- tainly arrive at a form (e„ -S„ €,_i), in which S,= ±1, and either e, = 2, e,_i = 4, or vice versd e,_i = 2, e, = 4 ; so that there are four cases (1,2) +J, = 2Z).-i-A, A=J,' + J,D,_, + 2D!.„ (3,4) +J, = 2A-A-i, A = J^ + J.D, + 2D!_,. Let us next suppose that n is even, so that ( — 1)"Z)„ = A is a positive determinant. Then it is evident S^, S^, ... are all positive, for 2» ^. = - i^nP, -Pn q.) (qnP.-l -Pn ?i-l) + ^2. 2.-H of which both parts are positive. Agaia, the numbers e^, e^, ... form a conti- nually decreasing series; for q„€, = [q„p,—p^q,y — Aq^ ; of which the positive part continually decreases, and the negative increases in absolute magnitude. But ei = g'„, and e„= — Ag„; there exists, therefore, a term e,_i which is positive, while the following term e, ia negative; whence ^^ = A-|-e,e,_i < A. Thus if A = 2, we shall have ^, = 1, e,_i = l, E,= -l, 2J, = D, + D,_„ A = {J. + D.y-2D',={J. + D._,Y-2D^_,. If A = 3, we shall have either (1) ^, = 1, e,_, = l, 6,= -2, 2J. = D, + 2D,.„ A = {J. + D,..,y-3D^.,; or (2) ^. = 1, e,_, = 2, e,= -l, 2J, = 2D. + D._„ A = {J. + D.y-3Dl K a is the integral number immediately inferior to VA, the period of integral quotients in the development of VA is of the type Ml, /*2. •••, Mife-i, b, ftk-i, fik-2, •••,Mi, 2a; and it is sometimes possible to assign d, priori the value of D^, the denominator of the complete quotient corresponding to b ; for that denominator is always a divisor oi 2A, and is besides <2\/A. Thus if ^ is a prime, -Dj = l or 2; if ^^ is a prime, -Djt = l, 2, or 4. Hence if ^ or Jyl is a prime of the form in + l, (-!)'•■ Z^j =- 1 ; for the equations x^-Ay'^= ±2, =+4 are impos- sible on the supposition that x and y are relatively prime, and the equation x^ — Ay^ = l is inadmissible, because 6 is not the last quotient of a period. Similarly if .4 or ^^ is a prime of the form 4m-f-3, (-l)''X>j = 2 or -2, according as the prime is of the form 8m + 7 or 8m + 3; i£ ^A is a prime of the form 4m + 3, (-l)*Z)i= -1-3 or -3, according as the prime is of the form 12m-fllorl2m + 7; and, in general, if X and — are each of them a prime of 288 REPORT ON THE THEORY OF NUMBERS. [Art. 123. the form 4n + 3, and if 2\<^A, ( — l)*Z)jt = X, or —X, according as X is or is A not a quadratic residue of — • We thus obtain a direct method for the repre- sentation of primes of the forms 4m + l, 8m + 3, 8m + 7, or the doubles of such primes, by the forms x^ + y% x^ + 2y\ x^ — 2y^: when X is a prime of the form 12m + 7, the developments of /^/JX and 2,^/^ will give representations of 3X by the forms a? — xy + y^, x^ + 3y^ : when X is a prime of one of the forms 287n+ll, 28 m + 15, 28m + 23, the development of ->/yX will give a repre- sentation of 7X by the form x^ — xy + 2y^, &c. The theorem relating to primes of the form 4/i + l is very celebrated; it was established independently by Gauss and Legendre, and it no doubt sug- gested the researches of Goepel in his doctoral dissertation 'De quibusdam aequationibus indeterminatis secundi gradlls' (Crelle, vol. xlv. pp. 1-13). Goepel confined his investigation to the case Z)„ = 2, though his method, which in the main is that here described, is of a much more general character. The theorems relating to the case A = — 3 were first given by M. Stem, who employs Goepel's method with very little modification (Crelle, vol. liii. pp. 87-98). A paper by M. Hermite, which appeared in Crelle's Journal (vol. xlv. p. 191) prior to the republication there of Goepel's dissertation, contains a method (see pp. 211-213) which is very similar to that of Goepel, but which does not connect itself so readily with the common theory of continued fractions. In these researches of M. Hermite the invariant ac' — 2bh' + ac appears expUcitly; which is not the case in Goepel's paper. EEPORT ON THE THEORY OF NUMBERS. Part VI. [Report of the British Association for 1865, pp. 322-375.] 124. Application of the Theory of ElUptic Functions to Quadratic Forms. — TTie Theta Functions of Jacohi. — It will be for the convenience of the reader to give in this place a brief statement of a few principles and results which belong to the theory of elliptic functions, and to which we shall have occasion to refer in the following articles. The Theta functions of Jacobi are defined by the equation m=—ao if e'''" = q, by the equation m= — » In these equations, fi and v are given integral numbers ; w is an imaginary constant, having for the coefficient of i in its imaginary part a quantity dif- ferent from zero and positive ; so that the analytical modulus of q is inferior to unity, and the series defining the Theta functions is convergent for all values of x real or imaginary ; lastly, a is a constant at present undetermined, but to which we shall hereafter assign a particular value depending on that of w. When it is not necessary to specify the value of <o, we shall write 6^ y (x), instead of 0^_ , (x, «). The following equations are immediate con- sequences of the definition of the Theta functions : pp 290 REPORT ON THE THEORY OF NUMBERS. [Art. 124. ^^2.A^) =(-1)" ^,A^)- e. li,v + 2 (^) e^,{x). ■'li + H,v + v (1) (2) (3) (4) • -^ (5) {x)= 0^,(x + i(/a) + /)a)xe"[''1+i''"'"-i'-'l . (6) Thus there are only four different Theta functions, ^o,o(^)j ^o,i(^). ^i.o(3j)> ^1.1(3^) (equations 1 and 2) ; of these, the first three are even functions, the last an uneven function (equation 3) ; they are all periodic, having a or 2 a for their period, according as /x is even or uneven (equation 4) ; the quotient 6^y {x) -i- 0^',,^ (x) is doubly periodic, having aw or 2 aw for its second period, according as v — / is even or uneven (equation 5) ; finally, any one of the four can be expressed as the product of any other by an exponential factor (equation 6). The identical equations 1 + q {v + v-^) + q*{v^ + v--)+q'>{v^ + v-^) + q^^{v* + v-*)+... = {l-q^){l-q*){l-q%. X (1+2^^) (l + q^v) (l + q^v)... x{l + qv-'^){l+q^v-')(l + q^v-^).... (7) (8) q* {v + v-^) + q^ {v^ + v-'^) + q^ {v^ + v-^) + ... = (1 - q^) (1 - q*) (1 - qo)... X qi (v + V-^) X {1 + q^v^) {1 +q*v^) {1 + q^v^)... X (1+92^-2) (l + g*'y-2)(l + g6v-2)... in which v is any quantity whatever, and q any quantity of which the ana- lytical modulus is inferior to unity, express an important property of the Theta functions. Elementary demonstrations of the first have been given by Jacobi and Cauchy* ; the second is immediately deducible from it, by writing * Jacobi, Fundamenta Nova, pp. 176-183 ; Crelle's Journal, vol. xxxvi. p. 76 ; Cauchy, Comptes Kendns, vol. xvii. pp. 523, 567. See also the note (by M. Hennite), ' Sur la Th6orie des Fonctions Elliptiques' in the 6th edition (Paris, 1862) of Lacroix, Traits Elementaire du Calcul Difforentiel, vol. ii. p. 397. iirx inx {In (7), v= e ' ; in (8), v = e " . Hence it would be better in (7) to write v' for v : then to obtain (8) by putting g^v for v, and multiplying by q^v. This transformation of (7) into (8) is immediately suggested by a comparison of imiitx HrX iwx ^..o(*) = 22"'e « , and e,_,{x) = Sg^' {qe ' ^.q^e » .} Art. 124.] REPORT ON THE THEORY OP NUMBERS. 291 qv^ for V, and multipljdng by qiv. We infer from these identities the four formulae * : + '" 2m-7rx eo,o(a:)j=53(— )} = 2^5«cos-^ 1 1 ^ a / cos 27ra; a 2m7ra; 27ra; (9) (10) = n„(i-22»)n^Ci_2g2m-icos-^+g*'»-2); .... 1 1 ^ * ei.o(a;)( = 3.(— )} = K.q^'""^'^' cos (2m + l) — i^J— QO ^ = 2qi cos — n„ (1 - 2'*») — n„ (l + 22="" cos ^ + q'A ; . . (11) lei.i(«^)(=3x(^)}=2Vl)'»5i(— )'sin(2m + l)'^ = 2^Bm — U„(l-q^^)U„(l-2q^^cos^^ + q*'");. . . .(12) ft 1 1 ^ Ob ' by which the Theta functions are expressed as convergent products of an in- finite number of factors. Other important consequences are deducible from the equation = ^<r-^,.a'-..(«-a;i) X 0<r-^.o'-.,(«-a^2) X d, _ ^3, ^. _ ,, (s - ^3) X 0,, _ ^, ^ _;,,(«- iK*) + ^CT-;ii,<;'-i'i + l('*~^i) X ^(r-fi2,<r'->'j + i(*~^2) X ^<r-/i3,<j'->'3 + l(* — ^3) X ^<r-/<«,ff'-i'4 + l(*~'^4) + (-l)'^0<,_^i+i.a'-Ki(s-a;i) X 0^_^+i.^_,,(s-X2) X d^_^3^.i.<^_,3(s-a;3) + (-l)i+<^0^_;,j+i.^_,.+i(s-a;i)x0^_^^,,^_,,+i(s-a;2) X aj_^3^.],5'_V3 + l(* — ajg) X 0ff_;X4 + l,<j'-I'i + l(*~"^4}> ^ * {This 5-notation is that employed by Jacobi in his Lectures; see Enneper, Sect. 15, p. 78. It gives 5,(3!) = 2 2™'' cos ima:. (See also p. 95.) The notation further gives iK(x) 1 S,(a;) sin am =^ = —r- —j-r • Perhaps it might be best to use it with double sufces. To these notations we must now add that of Glaisher's tables ; = 0, 0, = 4" H, 0, = ^ Hj, 0, = -/k' 0i.} ' Vk VIC P p 2 M13) 292 BEPORT ON THE THEORY OF NUMBERS. [Art. 124. which contains four independent arguments, Xi, X2, aJs, 0:4, and in which 2s = Xi + Xa + X3 + Xi, 2<r = Ati + /«2 + /*3 + M4. 2<r' = vi + ya + va + vt; the numbers /i^, ft^, n^, n^ and »'i, •'2, 1'a, •'4 being subject to the restriction that their sums are respectively even, so that o- and <r' are integral *. Let v/c, v «' be two quantities defined by the equations V /c = ^1.0(0) 00.0 (0)' attributing in (13) to the elements the systems of values v; 00.0 (0)' (14) Xi, Xa, ajj, «4 f^u M2. /*s, ^4 "i. "2. Vs, "4 0, 0, 0, (i) 0, 0, 0, 0, 0, 0, X, X, 0, (ii) 1, 1, 0, 1, 1, 0, X, X, 0, (iii) 0, 0, 0, 1, 1, 0, * This very symmetrical formula is, it would seem, nearly the same as that employed by Jacobi in his Lectures on Elliptic Functions at the University of Konigsberg (see his letter to M. Hermite in Crelle's Journal, vol. xxxii. p. 177). It may be proved by actually multiplying the four Theta series, irr and transforming the indices of —1, e "", and e" in the general term of the product by means of the elementary formulae a'+ft'+c'+d" = {8-af + {s-bf + {g-ef+{8-d)\ oa + 6/3 + cy + <Z8 = (s-a)(2-a) + (8-6)(S-^) + {8-c)(2-y) + (8-d)(S-8), where 2s = o + 5 + c + d, 22 = a + /3 + y+6. {Put (i) |'^ = '^ = A^ = M4=0, ,;;. f^=^, = ^ = ^.= l, ( r, = r, = r, = v, = ; ( 1;, = D, = v, = r, = ; and add ; we find n a„.„(9;<,)+'n 0^^,{x„)=''u 6,^,{b-x„) + 'u 0,^,{s-x„). O-sl ffj=l (r=:l o-=l This is given by Rosenhain as Jacobi's Fundamental Formula (M^moires dep Savants Etrangers, vol. xi. p. 61.)} [The formula (13) forms the subject of Professor Smith's paper 'On a formula for the multipli- cation of four Theta Functions, No, XVI.] Art. 124.] we obtain successively REPORT ON THE THEORY OF NUMBERS. 293 ' + /^ = 1, we find (15) ic'ei,(x) = ei,{x)+K6i^(x), (16) >c'eu(x)=di,(x)+Kei,(x) (17) Again, attributing to the same elements the values x + y, x-y, 0, , 1 , 1, 1 , 1 , 0, 0,, , (x - 3/) 00, i(x + y)6,, (0) 00. (0) = &i.iix)eo,r{x)eo,,{y)e,,,{y) -^o.o(a;)0i,o(«)e:.i(2/)0o,i(2/)- Dividing by y, and diminishing y without limit, we obtain d Ai(a;) \ ^ 0o,i(O)0'i.i(O) 01.0 (a;) 00.0 (a;) ..j,s dx\^{xy 0i.o(O)0o.o(O) 6l^{x) ^'°^ Similarly, we might form the differential coefficient of any other quotient of two Theta functions ; of these we require only the two following : <^ / ^i.o H \ _ ^0.0 (0) ^1,1 (0) ^1.1 ( ^) ^0.0 i^) MO ^ dx\e,_,{x))- d,,o{o)e,,,{o) ei,{x) ' ^'''''' d / Oo.o {x) \ _ 01. (0) 0;, 1 (0) 01. jx) 01. 1 jx) , .. dx\e,,,(x)) 00.0 (0) 00.1 (0) ei,(x) ^^^''^ We shall now attribute to a, which has hitherto been left indeterminate, the value 2K, K being a constant, the square root of which is determined by the equation '^^ = 0o.o(o)=i!r' = n„(i-2'"")(i+2^"'-?; • • • (19) V^ IT we shall also write K' for find from (10) and (11), '27k (oK i Attending to the values of V< and ^/k, we A/^^ = 0o,i(O) = 2l(-l)-2"'^ = n,„(l-2-)(l-2— ^)^ . (20) x/i!^ = 0,.o(O) = 2l2i<^»+i)^ = 2gm„(l-gr^™)(l+?«T • (21) * IT —00 1 Multiplying together the infinite products (19), (20), (21), and reducing by an identity of Euler's, n(i-2— o = n„^r-^, (E) l^l+g" 294 we obtain also V REPORT ON THE THEORY OF NUMBERS. [Art. 124. 2-'iS = ^2m„(i-r?-)3=^g(_i)™(2m+i)^<--^>' = ie;.,(o). (22) These equations (19-22) are of great importance in the arithmetical appli- cations of the theory. The constant a having the particular value 2 K, the ftmctions 1/ are denoted by Jacobi by the symbols 6i {x), 6 (a), H^ (a), H {x) ; we shall find it convenient occasionally to employ this notation. The elliptic functions (properly so called), sin am x, cos am x, A am x, are defined by the equations sm am 05 = — r- ^ ; : : cos am a; = —r- ', / ; A am a; = ^//c - ^ ; '■. . (23) Vk B [x) ^/k {x) ^ (^) These functions are all doubly periodic, having for their periods 4K, 2{K'; {2(K + iK')}; 2K, 4iK respectively; introducing them into the equations (16-18), we obtain cos^ am x + sin^ am x = l, A^ am x + k'^ sin^ am x = 1, (24) d . sin am x dx d . cos am x dx d . A am x dx = cos am X A am x, = — sin am X A am a;, = — K^sinamxcosama;. (25) From these formulae it appears that if y = sin am x, x is one of the values dy of the integral / ^ = . All the values of that integral are re- presented by the formula x + 4:mK + 2m'iK', in which m and m' represent any integral numbers whatever. Since sinamK = l, K is one of the values of the integral / — 7- ^ ; and it can be proved that K' is one of the values of the integral / ^ = . When the real part of w vanishes (in which case q, K, K', k, k are real and positive, and k, k less than unity), K and K' are the ordinary values of those definite integiuls ; i.e. the values Art. 125.] REPORT ON THE THEORY OF NUMBERS. 295 * obtained by causing y to pass from tbe inferior to the superior limit, through a series of real values. The well-known formulae of Addition and Subtraction which express the elliptic functions of the sum or difference of two arguments in terms of the elliptic functions of the arguments themselves, are easily deduced from (13). But as we shall not require these formulae in the following articles, we may omit them here. 125. The Modulus and its Complement. — The Theory of Transformation. — In the arithmetical application of the theory, the functions k and k, which are respectively termed the modulus of the elhptic functions, and the complement of the modulus are of primary importance. They are respectively fourth powers of the quantities oo 1 -I- O^"* "' 1 q2»1— 1 ^ = y2gin„. . J„_, ; ^^=n^ i^^..-i ; ..... (26) 1 l + ^^-n-l' *™l+g^"'-l' • ■ • • which are themselves perfectly determinate functions of «, if we understand the positive square root of 2 by a/ 2, and es'"" by qi. Of these functions, which we shall designate by (p ((e) and >\r («), the following equivalent expressions have been given by Jacobi (Crelle's Journal, vol. xxxvii. pp. 75-77) : \nj ^6f»^ + 2m U _ /g,,.n ^"g*" /2.7l ^(-l)"'g° - V"i* '■'■ l\j^qim-l\ (1 _o2m\ ~ V ''^'2(- l)2"'^"» + ^)02^3"»' + '»' ' \ 2 (i4m2 + 2n» 2 q;'^^ = -/2 01,0 (0,ia,)' {l-qi^-^y^l-q*"*) 2(-l)"'g2m2 _e-»''ei.o( 0,l(«+l)) - v/2 00,1 (0,2a,) ' y.(27) (0, !«,) «' = n n (1+2^ i-g" 1)2(1 -g^".) x/2 00,0 (0,») 2(-l)"'g2(3'»' + '») M '=n ft -.nl -9^"-')(l-<Z*™) 2(-l)"'r/"--^"'___,,^e,,o(0,l(a,+ l)) -f-«7'''"-')(l-g*'") 2:5''""'+' = e" 01,0 (0,i«) _g2m-1^2(l_g2«. ) _ S(-l)'»(7"''' _ 00,1 (0,0.) _ _ (^4^-2)2 (l_g4m) - 2 ( _ !)"• g2'"^ ~ 001 (0, 2 O)) ' -g*'»-^)^(l -(/"') _ 2(-l)"'g^'»' _ 00. 1(0, 2 co) + g,2m-lJ2Jl_g2n.j - 2 2"" ~ 00,0 (0,«) ■ y (28) w 1 296 REPORT ON THE THEORY OF NUMBERS. [Art. 125. These expressions of u and u' may be verified by a comparison of their general factors with the general factors in the formulae (26) : for some of them', this comparison requires the Eulerian identity already cited (E). Limits of 11 and 2 are 1, + oo , and — oo , + oo ; the transformation of the products into sums is efiected by means of (7). If ft) = a + hi, and if the positive quantity 6 increases without limit, a re- maining finite, we infer, from (26), that lim ■vl' (a + 6i) = + 1, lim — -— = cos iair + { sin Aax. We shall presently see that <^ (o)) = >//■ ( — j; hence if w = =- and 6 increase without limit, • lim^(-^) = limx|^(6i)= +1, lim x|^ (|-) = lim ^ (i6) = 0, lim— —^ = 1. The principal properties of ^ {te) and -^ (w) are deducible from the Theory of the Transformation of Elliptic Functions. The general problem considered in that theory is ' Given co = , ^ , where a, h, c, d are positive or negative in- tegral numbers, to express the Theta functions containing S2 by means of the Theta functions containing w.' The determinant ad — he must be different from zero and positive, because the coefficients of i in the imaginary parts of w and Q must both be different from zero and positive ; if ac? — &c = n, the transformation is said to be of order n. Let A, A', X, X', v, v be the same functions of Q that K, K', K, K, u, u' are of to; since Q = i^-, ft) = i-rr, the equation w = =-= A Jv ^ a + bil implies the existence of two others of the form jTj- K =aA-|-6iA', ^ 1 (29) -^iK' = cK + diA.';) in which M is a coefficient termed the multiplier ; when A has been found, M is determined by the equation l^ = :^{a + hQ) = ~{c + d^); (30) it also satisfies the relation * Fundamenta Nova, p. 75. Art. 125.] EEPORT ON THE THEORY OF NUMBERS. 297 If 7i= 1, the theory of the transformations of the first order has been com- prised by M. Hermite in the single formula *, in which r m = aiJ.+ bv + ab, n= Cfi + dv + cd, S />— i»ir(acu2 + <ibeav + bdv^ + 2ahea + 2ahdv + ab'^c) "m> " ^ "^ > = (— ) i'^", if a is uneven, or = (j-j i'z" X {- j(<»-i)(6-i)^ if J) is uneven f ; * Liouville, New Series, vol. iii. p. 26 ; and, with less detail, in the Comptes Kendus, vol. xlvi. p. 171. t These determinations of the value of / coincide with those given by M. Hermite in Liouville's Journal, vol. iii. p. 29 ; where, however, it would seem that the formulae relating to the two cases of ' a pair ' and ' a impair ' ought to be transposed. {Observe that, if we denote i-i^' ^^_„ by ^^_„, (32) acquires on the right-hand side the factor join-iii'j and £mn-iiv y g _. gjjir {aC|ii'+2ftcni'+6cJi''+2(K:dn+2ix:di'+oic(2d-6)|. Observe also that V — i(a + bil) = i-i A/a + bil, the real parts in both radicals being positive. It is convenient to divide by i"* ; so that J^ = iiJ= f — ^ i-4(<»-i), a uneven, = (-j-) i-l^f"-!), 6 uneven, 5,..([.+«] '^. a) = -^i^, X .-•<•"'» I . .... (f , .). or, since ^'*'' ("Ix ' ^) ~ ^'"•'' ^'^' ^^' putting h=. 2K, as in the text.} 0^,, (~, Q) = (7 X «~ «^^^ X e„.„ {x, a>), Q q 298 REPORT ON THE THEORY OP NUMBERS. [Art. 125. V the radical ^ — i{a + bii) represents that square root of —i{a + bQ), of which the real part is positive ; lastly, A is determiued by the equation 2^ = eo.„(O,Q) = -^=^^J==0„,,,(O,«), .... (33) which is a particular case of the formula (32) ; and M by the equation The formvda supposes that b is different from zero and positive ; if 6 = 0, we may suppose a = d = l, so that <t> = c + Q, and the formula of transformation is ^..(^>^) = e-i''"''e^,,,,,,{x,a,), (35) where 1^ = M4- The equations of the annexed Table, Avhich, for any transformation of the first order, express the relation subsisting between the given and the trans- formed modulus, are also due to M. Hermite, and are of great importance in the theory of the functions (p (w) and -^ (w) *. They may be obtained by apply- ing the formula of transformation (32) to the expressions of (p (w) given by Jacobi (27). There are six cases, answering to the six solutions, of which the congi-uence ad — bc = l, mod 2 is susceptible. We add, in each case, the value of the multiplier f. * ' S»ir la resolution de I'^quation du cinquifeme degr^,' Comptes Kendus, vol. xlvi. p. 608 ; or in a separate reprint (including other memoirs from vols. xlvi. and xlviii.) with the title ' Sur la thfeorie des Equations modulaires, et la resolution de I'^quation du cinqui^me degr6,' p. 4. + [The column giving the transformations of yfr («)) was added in manuscript by Professor Smith. He mentions that the values in this column were taken from Koenigsberger, Clebsch's Annalen, vol. iii. p. 10, and verified by The Bubject-matter of §§ 124 and 125 is considered in much greater detail by Professor Smith in his 'Memoir on the Theta and Omega Functions,' on which he was engaged at the time of his death.] Art. 125.] RKPOET ON THE THEORY OE NUMBERS, Table A. 299 ad— be = 1, a- = e5« a = bB c = d= *(«) = n<-) = 1 M ~ I. I 1 (1) <r''<t>{^) (i).--v'(«) (_l)i(<>-i) II. 1 1 (|)cr-<^V(i2) (|) -"-^C^) (_l)4(f-i)«- III. 1 1 1 ^d^ <f>{il) 4, (a) (_l)iCo + o-l)^ IV. 1 1 1 (_l)i(«+i),\ V. 1 1 1 (_l)i(a-l)/ VI. 1 1 1 (-l)*^-!)^/ It would be easy to write these equations so as to express ^ {Q) in terms of (p (<o), thus completing the solution of the Problem of Transformation of the first order ; but it is more convenient to retain them in their actual form. Similar formulae exist expressing >[' lee), ,) { , in terms of (p (Q) and ^ (Q) *. The propositions implied in the equations of the Table may also be enun- ciated conversely. Thus to case I. corresponds the theorem ' If w and Q are imaginaries in which the coefficient of i is positive, and if ^^'' (o) = ^2" (Q), four integral numbers a, b, c, d can be found satisfying the relations G + dQ a + 1)^1 ; ad-hc = l; a = d = l, mod2; & = 0, mod2; c = 0, mod 2*-'.' * M. Hermite has also shown that the function X(«) = 4/2.5^(1-3) (l + 2')(l-?')(l+?0-. . ., e + dil , , which is a cuhe root of </>(a)) X >/'(«>). possesses a similar property; viz. if w = , ,^ > ad — bc=\, \{a>) can be expressed in terms of x(P), </> (i^), and \/^ (ii). (Sur la th6orie des Equations Mo- dalaires, p. 15.) Qq2 300 KEPORT ON THE THEORY OF NTTMBERS. [Art. 125. If (f){a)) = <j) (Q), four integral numbers a, h, c, d can be found satisfying the relations w = j-a > <!i^ — bc = l; 6 = 0, mod 2 ; and either a = d= ±1, mod 8 ; c = 0, mod 16, or a = d= +3, mod 8, c = 8, mod 16.' These converse propositions may be demonstrated by means of the diflPer- ential equations satisfied by the eUiptic functions ; by a similar process we obtain the following equally important theorem : — ' If ^ is any quantity, real or imaginary, other than zero or positive unity, there exist values of w, having the coefficient of i in their imaginary parts different from zero and positive, which satisfy the equation (p^ («) = A.' When w is an uneven integer other than 1, the formula of tranformation is e,,.(~,Q) = Te-^-^e„,,{x,a>), (36) in which w and n are determined as before, and T is a homogeneous function of order ^(/i — 1) of the squares of two of the functions 0^y{x, w). We need not occupy ourselves here with the determination of A and T, but shall confine ourselves to the consideration of the modulus and multiplier alone. Represent- ing by $ (n) the sum of the divisors of n, every binary matrix of order n is ' = I A I X I e |, in which | c | is an unit matrix, and | A | included in the formula one of the 4> [n) matrices , 7 and y being conjugate divisors of n, and k c, d 7,0 ky representing any term of a complete system of residues, mod y. It is thus sufficient to consider a system of 4> (n) transformations of order n, since all others arise from compounding transformations of the first order with the transforma- tions of that system. If we take, in particular, the system of transformations, 7,0 -16k + y'Q A- ^ ^r. — , correspondmg to the matrices -16^-, y (since n, and there- 7 fore 7' is uneven, we may take a system of residues, mod 7', of which every term is divisible by 16), we have for the determination of the transformed modulus, the fundamental theorem *, * M. Hermite, Sur la tWorie des Equations Modulaires, p. 36 ; M. Joubert, Comptes Rendos, vol.1, p. 774; or, in a separate reprint with the title 'Sur la Th6orie des Fonctions Elliptiques, et son application a la Th(k)rie des Nombres,' p. 21. The demonstration o." this theorem for the case in which w is a prime, is contained in Sohnke's important memoir ' jEquationes modulares pro transformatione functionum ellipticarura,' Crelle, vol. xvi. p. 97. From this particular case, the truth of the theorem for any value of n is inferred without difficulty. Art. 125.] EEPORT OK THE THEOBY OF NUMBERS. 301 ' The quantities (-) <p (Q) = (-) <^ (^^^ ) are the roots of an equation of order $ {n), in which the first coefficient is unity, and the other coefficients are rational and integral functions of cp (w) having integral coefficients.' This equation is termed the modular equation of the transformation of the ?ith order; designating <^ (a,) by ii, and (^-'^(j)(T^^ — ^ by v, we shaU re- present it by /(«., tt, v) = 0, or more simply by f{u, v) = 0, The function f{u, v) is characterized by the following, among many other properties, (37) /(«, V) = (-!)*'») n (?) X (u^,)*(»)/(l, I) . If in the equation /(«, v) = we put « = \/^ (w), the roots are represented by If we put u^ei*" cf){w), where s is any integral number, the roots are re- presented by ^2^ ^,^^^^ ^ ^ ya, + 16k ^ ^ If u = e*'", there are y' roots represented by (-jei*''"', y denoting any divisor of n. If we put u = e^'" , ) { , where s is any uneven number, the roots are represented by ^^-y«-H6/fc> * 7 The equations whose roots are respectively the squares, fourth powers, and eighth powers of the roots oifiii, v) = 0, contain only the squares, fourth powers, and eighth powers of u ; we shall represent these modular equations by /, (u^ V') = 0, / {u\ V') = 0, or /, {k, X) = 0, and /« {u\ v^) = 0, or f, (<c^ X^) = 0. * It is easily seen that v = <f> (co) is one of the roots of / (—7) </> ( ? )> v\ = 0: this establishes the first of the equations (37). The other properties given in the text are deducible from the equations M = (^(ft)), v = (-^ <l>( , ), by applying to to different transformations of the first order, and employing the formulae of the Table A. 302 KEPOET ON THE THEORY OF NUMBERS. [Art. 125. The last equation (by what has preceded) remains unchanged if we write (1) K for X, and vice versd, (2) 1 — k^ for k^, 1 — X^ for X^, (3) - for k, -■ for X. K X K n admits of a square divisor S^, f{n, u, v) is divisible by / (-^ , u, {^) '^jl for if yy'=~, v = ^_)^^!yfL_ — ) is a root of /(^, u, v) = 0, and W/ ^ V^ — S~' / " V^/ ^ ^ * of /(». ^> ^) = 0. It is sometimes convenient to suppose that the modular equation has been freed by divison from the factors corresponding to the quotients of n divided by its square divisors ; its degree will then be i£ Pi, P2, ... represent the primes whose squares divide n, or «n (I4--), if ^ represent any prime dividing n. The roots of this reduced modular equation are expressed by the same formula as before ; only that y, y, and h are now subject to the condition that they must not have any common divisor. With regard to transformations of an even order, we shall only have occasion to consider the case in which n is a power of 2. If n = 2, we have the modular equations, 2 m^ 1 — u* f* = 7-; — T, v* = - -, (38) 1+M* 1+u* ^ of which, i£ u = (f) (w), the roots are given by the equations If we represent the modular equation of order 2*^, when cleared of fractions, by f{2i',n,v^) = 0, the modular equation of order 2''+i, or /(2'' + i, w, v^^.i) = 0, is obtained by eliminating v^ from the two equations f{2^u,v,) = 0, and v*,,,= ^^- We may thus successively calculate the modular equations of the orders 4, 8, 16, ...; and, attending to the expression, by means of the transcendent (p, of the roots of the equation v* = i-^— 4 , we may establish the following proper- ties : — the function f{2'', u, v^ is of the order 2''-2 in v', or of the order 2** in m^ ; the coefficient of t*^''"^^ is v^*^*^, and the equation is not altered by writing Art. 126.] REPORT ON THE THEORY OF NUMBERS. 303 — for u^ and multiplying by m^mi-i. y u = (p{w), the values of v are given by the equation V^ = ^8 t- (39) in which k represents any term of a complete system of residues, mod 2''-^, and correspond to the transformations defined by the formula c + dQ a, b c, d 1,0 -8^,2'' 1,0 2h, 1 a + bQ' where h is any term of a system of residues, mod 8 ; if •?; = ^ (Q), the values of u are given by the equation / 2''Q \ ""'^'dms) (*») where h is any term of a system of residues, mod 2^ For the determination of the multiplier in a transformation of an uneven order n, we have the theorem, ' If M is the multiplier corresponding to the transformation w = — , , 1 . ^ . the $(ft) quantities z = ( — l)?'"'"''^ satisfy an equation of order $ (n), in which the coefficient of the highest power of z is unity, and the coefficients of the other powers of z are rational and integral functions with integral coefficients of K^ ; the absolute term, in particular, being + « ' *. 126. The Comjilex Multiplication of the Argument. — The problem of the multiplication of the argument is 'Given an integral number n, to express the Theta functions of nx and w by means of the Theta functions of x and w.' The solution of this problem may be made to depend on that of the addition of arguments ; for to add n equal arguments is to multiply the argument by n. The problem is also included in that of transformation ; for if we consider n, the transformation of order n'^, of which the matrix is 1 0, n , we have i2 = co, A = K, A' = K', ^ = n When ft) is not the root of a quadratic equation having integral coefficients, * Jacobi in Crelle's Journal, vol. iii. p. 308. M. Joubert (Comptes Rendus, vol. xlvii. p. 341) has calculated the equations of the multiplier for the orders 3, 5, 7, 11. See also M. Brioschi in Tortolini's Annals, vol. i. (New Series) p. 175, M. Hennite, Equations Modulaires, pp. 12 and 31. No complete demonstration of the theorem appears to have been given. 304 REPORT ON THE THEORY OF NUMBERS. [Art. 126. n, 0, n the transformations, of any square order «.*, and of the type transformations which do jiot alter the value of «. For if w = are the only c + dQ . a + bii to = Q, we have bu)' + (a — d)u) — c = 0. But, by hypothesis, a> is not the root of any quadratic equation having integral coefficients ; neither is w rational ; therefore the three numbers b, a — d, and c are all zero, and the matrix a, h c, d is of the type n, 0, n But if w is the root of a quadratic equation hav- ing integral coefficients, an infinite number of transformations, other than those included in the formula n, 0, n , can be assigned, which do not alter the value of o». Let A + 2Bw+ Cw^ = be the equation satisfied by w ; and let ^ C — ^'^ = A ; then A is difierent from zero and positive ; also A and C are of the same sign, and may be supposed to be positive, so that w = j^ ; lastly, let 6 = 1, or = 2, according as {A, B, C) is properly or improperly primitive. Let n be any number such that d^n admits of representation by (1, 0, A) ; and let a, r be the values of the indeterminates in any such representation ; then the transformation U-+-^)' e 6 {<T-rB) and will tA ' d ' e of order n will not alter the value of w, because w = ~ ^' have for the reciprocal of its multiplier - [o- + T J5 + T C«] = ^ (o- + IT ^/ A). The transformations derived from different values of n, or from different repre- sentations of the same value, are all different ; and every transformation of order n which does not alter the value of w, is derived from some representation of 0^71 by (1, 0, A) ; so that the transformations and representations correspond to one another one by one, and are equal in number. It will be observed that the multiplier corresponding to any of these transformations is a complex factor (composed with ^ — A) of the number expressing the order of the transforma- tion ; so that the transformation is equivalent to a complex multiplication of the argument. And the Theta functions containing w do, or do not, admit of com- plex multiplication, according as w is or is not a quadratic surd. Art. 126.] REPOET ON THE THEORY OF NUMBERS. 305 If we consider the values of w contained in Theta functions admitting of Imultiplication with i %/ A, we see that these values are infinite in number ; each jform of determinant — A supplying one. But the values of (p^ (w), corresponding ito these values, are finite in number, being six times as many as the classes of forms of det. — A ; provided that in the enumeration of the classes a class of [det. — 1, or a class derived from a class of det. — 1, is counted as \ instead of 1 ; Fand an improperly primitive class of det. —3, or a class derived from such a ^class, is counted as \ instead of 1. For it appears from the Table (A) that the values of ^* (to) corresponding to two equivalent forms, are equal or not, accord- ling as the transformation, by which one form passes into the other, is or is not jf the type ' , mod 2. We have therefore only to ascertain how many U, X. ibclasses each class contains, a subclass consisting of forms equivalent by 1,0 Itransformations of the type 0,1 A simple discussion shows that the number ■of subclasses is six (corresponding to the six types of binary matrices for the modulus 2) ; except in the two cases just referred to, when the number of subclasses is reduced to 3 and 2 respectively, owing to the existence in those two cases of automorphics which are not of the type 0, 1 , mod 2. Thus the whole number of values of </)* («) is 6 (x (A), G (A) representing the number of classes of det. — A, counted in the manner stated above *. It will be seen that the six values of ^' (w) corresponding to the forms of the same class are of the 1 1 k'^ — Ik^ type /c^, — , , 1 — K^ (being in fact related to one another as the six anharmonic ratios of four points). The three values corresponding to the forms of det. — 1 are — 1, 2, ^ ; and the two values corresponding to the im- properly primitive forms of det. — 3 are the imaginary cube roots of — 1. It is an important theorem (to which we shall again refer) that the 6 G (A) values of <^' (&>) satisfy an equation of that order, of which the coefiicients are integral numbers (but the first coefficient not, in general, unity). The whole number of values of (p (w), corresponding to the forms of deter- minant - A, is 48 G^ (A). For if a be the value of ^ (w) corresponding to any form of a given subclass, and rj be any eighth root of unity, j/a will be a value of <^ («) corresponding to another form of the same subclass. * G (A) is the sum of the densities of the classes of det. —A; the density of a class, according to the definition of Eisenstein, being the reciprocal of the number of its automorphics. R r 2qi + 2q'i + 2q" + ..., 306 REPORT ON THK THEORY OF NUMBERS. [Art. 127. 127. Jacobi's Formulae for the number of decompositions of a number into squares. — The first applications of elliptic formulae to the theory of numbers were made by Jacobi. The developments, in series proceeding by powers of q, of the squares, fourth, sixth, and eighth powers of the functions '2K '2^ TT which are found in the ' Fundamenta Nova ' (sections 40-42, and 65, 66), are the analytical expression of arithmetical propositions relating to the composition of numbers by the addition of two, four, six, and eight squares. In these developments n represents any number from 1 to oo, v any uneven number from 1 to 00 ; c? is any divisor of n, S any uneven divisor of n or any divisor of c ; d' and S' are the divisors conjugate to d and S ; and the summations indicated by 2„, 2y, 2^, and 2 extend to every value of n, v, d, and S respectively. (I) y<.,+4S.(-l)!.-..ji:^.l+4S.j^. = l + 42 28(-l)i«-i)(/. (2) ^^-.,^,^.,r->^.-*^.^ -4S, S,(-l)il'-')j!'. (3) ^■ = l + 8S.-,^£^..l + 82, «- -- l+(_l)»g« »(1+(_1)»^»)2 = 1 + 242„ Zs Sq^ - 162, Z^^" = 1 + 8 [2 + ( - 1)»] 2„ 2j Sq\ m *''^'-162 *"?" -162 I'i^+'i"') = 162,28^/. (5) ^ = i + i62„-^^-42,(-lp-»)-'-^ = 1 + 4 2„ 25 ( - l)i(«-i) (45'2 - S') q\ (6) ?^ = 42,f-2^ — 42,(-l)i(''-')^^^^ ^ ' TT^ "^H-g' '^ ^ l-q" = 42,2j[(-l)i(»'-i)-(-l)i(»-i)]52g^. Art. 127.] REPORT ON THE THEORY OF NUMBERS. 307 16 K* n^n" (7) 1^=1 + 162. j-il^ = l + 162„2<j(-l)» + ''c?3^». <8) 1^ = 256 2:„^^„ = 256 2„2^3 22»_ Of these formulae, the first two are the analytical expression of the prin- cipal theorems relating to the composition of numbers by the addition of two squares (see Art. 95 of this Report) ; the others may be paraphrased as follows *. (3) 'The number of representations of any number iV as a sum of four squares is eight times the sum of its divisors if iV" is uneven, twenty-four times the sum of its uneven divisors if iV^ is even.' (4) ' The number of compositions of the quadruple of any uneven number N by the addition of four uneven squares is equal to the sum of the divisors of iVT.' (5) ' The number of representations of any number iV^ as a sum of six squares is 4 2 ( — l)J^'~i' (4^^ — ^2), S denoting any uneven divisor of N, S' its conjugate divisor. In particular if iV=l, mod 4, the number of represen- tations is 12 2(-l)§(»-i); ifiV=-l, mod4, itis -202 (- l)2(s-i>^2 ' (6) ' The number of compositions of the double of any uneven number N by the addition of six uneven squares is Je2[(-l)J«'-i)-(-l)|(«-i)]^- if N=l, mod 4, this number is zero ; if N= — 1, mod 4, it is _| 2 (_ 1)1(8-1)^2/ (7) ' The number of representations of any uneven number as a sum of eight squares is sixteen times the sum of the cubes of its divisors ; for an even number it is sixteen times the excess of the cubes of the even divisors above the cubes of the uneven divisors.' (8) ' If iV" is any number whatever, the number of compositions of 8 iV by the addition of eight uneven squares is equal to the sum of the cubes of those divisors of N whose conjugates are uneven.' * The expansions of (1) x (2), (1) x (4), (3) x (2), (3) x (4), are also given in sections 40 and 41 of the ' Fundamenta ' ; and may be similarly interpreted. R r 2 308 REPORT ON THE THEORY OF NUMBERS. [Art. 127. In counting the number of compositions by addition of squares, two com- positions are to be considered as different if, and only if, the same places in each are not occupied by the same squares ; but in counting the number of representations we have to attend also to the signs of the roots of the squares. Thus each composition by the addition of four squares, none of which is zero, is equivalent to sixteen representations. Only one or two of the preceding theorems are enunciated in the published writings of Jacobi : see Crelle's Journal, vol. iii. p. 191 ; vol. xii. p. 167. Some of the others have been given by Eisenstein (Crelle, vol. xxxv. p. 135), who had also obtained purely arithmetical demonstrations of them from the theory of quadratic forms con- taining several indeterminates. ' In my investigations,' he says ' these theorems are proved by purely arithmetical considerations, and appear as special cases of more general theorems ; at the same time we see why these developments close with the eighth power ; since, in fact, eight is the greatest number of indeterminates for which only one class of forms, represented by a sum of squares, appertains to the determinant — 1.' In the second of the notes to which we have just referred (Crelle, vol. xii. p. 167), Jacobi has given an arithmetical demonstration of the theorem (4). It consists in a kind of translation of the analytical proof into an arith- metical one ; and is of great interest and importance, as the first example of a new method, and as having suggested important researches to MM. Liouville and Kronecker (see Liouville's Journal, New Series, vol. vii. p. 48 ; M. Kro- necker, ' Monatsberichte,' May 26, 1862, p. 307). The doubly periodic functions of argument obtained by dividing any Theta function by any other, or the product of any two of them, by the pro- duct of the other two, admit of development in series proceeding by sines or cosines of multiples of the argument x. These developments, which, unlike the developments of the Theta functions themselves, are not convergent for all values of x, real or imaginary, will be foimd for the most part in section 39 of the ' Fundamenta Nova ' ; and the complete system has been given by M. Hermite (Comptes Rendus, July 7, 1862). One, which we require in this place, will serve as an example of the rest, kK . 2Ka _, qi'siiivx \ — sm^-- = X^^-^-^ (^^ = 2, 2j sin Sx .qi". ) It is from these developments that the expansions (1) ... (8) of the powers Art. 127.] REPORT ON THE THEORY OF NUMBERS. 309 of 2K -^ are deduced. Thus, writing ^ x for a; in (A), we find, since sin am K = 1, ^^ = Z.(-1).^-.^ 2" which is the fotmula (2). We shall now show how the equation (4) can be obtained .by squaring this formula. For this purpose we represent by a and ^ any two unequal positive uneven numbers congruous to one another for the modulus 4, and by a and /S' any two positive uneven numbers not congruous to one another for the modulus 4. We then have 4 ^2 ^y + 2„2, Ma + P) •-2„'2« yhW + ff) (1 - q'Y ' "" "'^ (1 - g°) (1 - q^) ""' "^ (1 - g«') (1 - q^) ' = P+Q-R, for brevity. Here P = 2. ^-^^^ = 2„ . 2^'. 2" = S« j^n ; again in Q, if we double each term we may suppose j8>a; let j8 = a + 4n; observing that a may be any positive uneven number, and n any positive number whatever, we find ^in + v ^ = 22„2, (1 - 2') (1 -?*""''') =22 2 g'" r g" g'""" 1 I I 4»— 1 = 22„ 2„ ■fin + v l" (l-g^Kl-g") liOstly, in i2 let a' 4- /8' = 4ft ; so that q2n 4»-l 22= 2, 2, 7 (i-gKi-g*"-") 4»— 1 n^n _ V V _g _g 4._g +1 4n-l 22,. 2, r (i-g^Ki-gO "i-g*" Consequently i — j- = P+Q-R which is the formula (4). 310 REPORT ON THE THEORY OF NUMBERS. [ah. 127. Thus by a purely analytical process we deduce from an equation which exhibits the number of compositions of the double of an uneven number by the addition of two uneven squares, an equation exhibiting the number of com- positions of the quadruple of an uneven number by the addition of four uneven squares. This analysis Jacobi has expressed arithmetically as follows. Re- presenting by iV an uneven number, by [4iV^] the number of compositions of 4iV^ by the addition of four uneven squares, we resolve 4iV^ in every possible way into two unevenly even numbers 2Ni and 2N2, and each of these in every possible way into two uneven squares ; we thus obtain the equation [iN] = i:[2N, = {2x+iy + {2y + iy]x[2N2 = {2x + iy + {2y + iyi in which the summation extends to every pair of uneven numbers iVj and N2 which satisfy the equation 2iV=iVi-|-iV2, and the square brackets represent the number of solutions in positive integers of the equations included in them. Observing that [2Ni = {2x + iy + (2y + iy] is the excess of the number of divisors of N^ which are of the form 4^-f-l, above the number of its divisors which are of the form 4^ — 1, retaining the signification of a, /8, a, /3', and denoting by a and h any positive uneven numbers, we may transform the expression of [4iV] into the following, [^N] = [2N={a + b)a] + [2N=aa + bl3]-[2N=aa+hl3^, in which the square brackets still retain the same signification. Supposing, as before, (3> a, and jS = a + 4 n, we have [2N = aa + bl3] = 2[2N = a{a + b) + 4:nh]; or, putting a = v -|- 4 kn, v being less than 4 n, [2N=aa + b^'\ = 2[2N=v{a + b) + ^n(^i + hk + b)'\ = 2[N=vx + 2ny], y being uneven and v <2n. Again, if in \2N=aa +b^''\ we write 4m for a + j8', and suppose a>b (the supposition a = 6 is inadmissible as it would render N even), we have {2N=aa: + b^'] = 2[N=a.^{a-h) + 2nb'\ = 2[N=vx + 2ny1, as before. Hence [2iV = aa + 6(8]-[2iV^ = aa' + 6|8'] = 0, and [4iV] = [2iV=(a-|-&)a], i.e. [4iV] is the sum of the divisors of N. In this arithmetical process we determine the coeflBcient of q^ in P, Q, R, instead of determining those functions 2wo^" themselves ; and as the difference Q — R= — 2„ - — ^-— is au even function in "l-gr*" the analytical process, so the difference [2 iV= aa -|- 6 ;Q] - [2 iV= a a' + 6/3'] vanishes in the arithmetical one. Art. 128.] REPORT ON THE THEORY OF NUMBERS. 811 Lejeune Dirichlet, in a letter addiessed to M. Liouville (Liouville's Journal, New Series, vol. i. p. 210), has put Jacobi's demonstration into a form in which it is more easily followed, but is a little further removed from the analysis. He shows that to every solution of the equation aa + h^ = 2N, in which a>(8, lere corresponds a solution of the equation a'a +h'^' = 2N, in which a'>/3', ' and vice versd, the two solutions being connected by the relation a, b' ^', - a x + 1, x+1 X , x+1 h, a X a, ■ -^ irhere x is the integral number immediately inferior to same thing, to ^' a-^' or, which is the Hence, as before, [2iV'=aa + &,S] = [2iV=aV + 6'/3'], and [4iV] is equal to the sum of the divisors of iV. 128. Theorems of Jacobi on Simultaneotis Quadratic Forms. — In an elabo- rate memoir ' On Series whose Exponents are of two Quadratic forms ' *, Jacobi has established a great number of elliptic formulae, which are the ana- lytical expression of theorems relating to the representation of numbers by certain quadratic forms. A comparison of the two criteria of Gauss for the biquadratic character of 2 with respect to a prime p of the linear form 8^-1-1, leads to a result which will serve as an example of these theorems. By the first criterion, 2 is or is not a biquadratic residue of a prime p of the form Sk-\-\ according as a is even or uneven in the equation ^ = (4a + l)^-f-86^ ; by the second, 2 is or is not a biquadratic residue of p according as /S is even or uneven in the equation p = (4a-f-l)2-fl6/32-f-. We infer therefore that a + ^ * Crelle's Journal, vol. xxxvii. pp. 61 and 221 ; or Mathematische Werke, vol. ii. p. 67. t Theoria Residuorum Biquadraticorum, Arts. 13-21. To the second criterion we have already referred in this Report (Art. 24, and in the additions to Part I., printed at the end of Part II.); the first is more elementary, and is inferred from the equation js = (4a+ l)'+86', in which /) is a prime of the form Sk+1. Raising each side of the congruence — 86' = (4a+lf, mod p, to the power \{p—\), and observing that 2iC»>->) = (-!)=!, (_i)i(P-i) = i, find ^^--HjX^y But if 6=2^/3, where /3 is uneven. (}) = (7) = (|) = '- because p = {4a + 1)', mod ^ ; and , 4a+]_. ^ ._p_. ^ ,_86^. ^ ,^_x ^ ^ p ^ V4a+1/ V4a+1/ ^'la+l^ ^ ' Hence 2»('~') = (—1)*, modp, which is Gauss's first criterion. 312 REPORT ON THE THEORY OF NUMBERS. [Art. 128. is even, or (since a + b + a is even by virtue of the congruence (4a + l)* + 8&2 = (4a + l)2, mod 16) that a + /3 + 6 is even. The result is thus generalized by Jacobi : 'For any number P the sum 2( — 1)*, i.e., the excess of the number of solutions of the equation P = (4 a + 1)^ + 86^ in which 6 is even above the number of solutions in which h is uneven, is equal to the sum 2( — 1)«+^, i.e. to the excess of the number of solutions of the equation P = (4a + 1)^ + 16/3*, in which a + /8 is even above the number of solutions in which a + j8 is uneven.' The generalized theorem is expressed analytically by the equation 2;( — l)» o(*'» + i)' + 8»2 = 2( — !)"• + " o^*"'^^'''*""'', (2) in which the simimations extend to all values of m and n from — oo to + oo . But this equation is an elliptic formula ; for, on dividing by q, and writing q for q*, it becomes 2 ( - !)» 5»' X 222«.' + m = 2 ( - 1)» q^»' 2 ( - l)" g2'»' + « which is included in the equations (28) of Art. 125, and is therefore a corol- lary from the fundamental property of the Theta functions expressed in equa- tion (7) of Art. 124. We infer at the same time, from the equations (28), that either of the sums SC — 1)"»+" 0^*"' + 1)^ + 18 »' or 2 (— 1")™ o^**""*"*^' ■'■*"'' is equal to the infinite product m = oo ' gn(i-28'»)(i-5""'). m = l We thus arrive at an analytical proof of Jacobi's theorem, including, as a particular case, a proof of the identity of Gauss's two criteria. But the con- tinuation of Jacobi's memoir was intended to contain direct arithmetical demonstrations (which, however, have never been published) of the theorems of which the equation 2( — 1)* = 2( — 1)"+^ is an example. He says, 'Though these arithmetical demonstrations of results obtained analytically present no essential difiiculty, yet they are sometimes of a complicated character, and require peculiar classifications of numbers which perhaps may be of use in other researches. We have here a certain amount of freedom in the choice of methods, so that the proofs can easily be varied'*. Probably one of these methods was that employed by Dirichlet in his earliest arithmetical memoir, * Mathematische Werke, toI. ii. p. 73. IArt. 128.] REPORT ON THE THEORY OF NUMBERS. 313 to which Jacob! expressly refers. In this memoir* (written when only the enunciations of Gauss's criteria for the biquadratic character of 2 had been published) Dirichlet gives a demonstration of the first criterion, which does not differ from that subsequently given by Gauss (Theor. Res. Biq. Comm. prima, Art. 13), and then deduces the second criterion, as follows, from the first. [Since rp = (4^ + 1)2 + 852 = (4„ + 1)2 + 16^2^ Iwehave . [4(a + ;8) + l] x [4(a-;8) + l] = (4a + l)2- 861 'No common divisor of 4 a + 1 and h can also be a common divisor of 4(a + ;8) + l and 4(a-(8) + l, \i.e. of 4a + 1 and h; for p is not divisible by any square. The greatest common idivisor of (4a + l)2 and h'^ must therefore be a product of two relatively prime leven squares ^^ and h"^, dividing 4(a + ,8) + l and 4(a-/3) + l respectively; ' -Yi is thus a divisor of the quadratic form a;^ — 8 if, in which x and y are relatively prime ; it is, consequently, itself of that quadratic form, and 4(a + ,8) + l = l, mod 8; this congruence implies that a + fi = 0, mod 2, or, which comes to the same thing, that b = a + ^, mod 2. It will be seen that this demonstration of the congruence 6 = a + /3, mod 2, applies to any two representations of any number P by the forms /= (4a + 1)^ + 8 fe^ and <^ = (4a + l)2 + 16/32, provided that in the two representations the four numbers 4a + l, 4a + l, b, /3 have no common divisor. To prove, for every uneven value of P, the truth of Jacobi's equation Z ( - 1)" = 2 ( - 1)" + ^, we observe, first of all, that the equation is evidently true if P is not = 1, mod 8, or if P contain an uneven power of a prime of the linear form 8k + 7 ; for in these cases there are no representations of P by either form. We may therefore suppose that P is of the linear form Sk + 1 ; then the equation is true if P contains an uneven power of any prime p of either of the linear forms 8^ + 3 ; thus if P=p^''*^P', where P' is prime to p, and p = P'=Z, mod 8, there are no representations of P by <^, so that 2 ( - 1)«+^ = ; let the equations p2K + l ^2.2 + 27/2, F = X'^ + 2Y^ denote generally those representations of p^'' + '^ and P' by the form (1, 0, 2) * Crelle's Journal, vol. iii. p. 35. S S 314 REPORT ON THE THEORY OF NUMBERS. [Art, 128- in which the first indeterminate is = 1, mod 4 ; then the representations of Pszp^' + ixP' by/ will be comprised in the formula P = {2yY-xXy + 8{^(i/X + xY)}^*; but of the two numbers ^(yX+xY), ^(yX-xY) (both being values of the second indeterminate), one is uneven and the other even ; whence S(-l)* = = 2(-l)«+^. Similarly if P=p^'' + ^P', where P' is prime to p, and p = P'=5, mod 8, there are no representations of P by/, and it may be shown that 2(-l)«+^ = = 2(-l/. We may therefore confine ourselves to the case in which P is composed of any powers of primes of the linear form 8^ + 1, and of even powers of primes of the forms 8^ + 3, 5, 7. If, on this supposition, P = P'xP", where P' and P" are relatively prime, and each is = 1, mod 8, the sums 2(-l)'' and 2(-l)"+^, relative to P, are the products of the corresponding sums relative to P' and P". This may be proved by observing that the representations of P by / [or <^] may be obtained by compounding the representations of P' and P" by that form, and that each representation of P has the character of an even or uneven h [or a + 18] according as the representations of P' and P" of which it is com- pounded agree or differ in respect of that character. Thus it is sufficient to consider the four cases in which (1) P=p'', ^ = 1, modS; (2) P=p''', p = 3, mod 8; (3) P^p^", p = 5, mod 8; (4) P=p^'', p = 7, mod 8. In the last of these cases it is evident that 2(-l)^=+l = 2(-l)"+^; in the others, the proof is supplied by Dirichlet's method. (i) If P^p", p=l, mod 8, there are two primitive and v — 1 derived representations of P by each form ; and the application of Dirichlet's method shows that, for every representation of P by cp, (-1)°+^ has the same value as (-1)* in either primitive representation of P by / and, conversely, that for every representation of P by /, ( - 1)* has the same value as ( — 1)°+^ in either primitive representation of P by <^ ; whence the units ( - 1)* and ( — 1)°+^ have all the same value, and 2{-l)*==2(-l)«+^=±(. + l). * For 2i/¥—xX = 1, mod 4 ; and the representations comprised in the formula are all different, their number being equal to the number of sets of representations of /• by (1, 0, 8). Art. 128.] REPORT ON THE THEORY OF NUMBERS. 315 The ambiguous sign is that of ( — 1)* in the primitive representation of P ^J f, and will be found (by reasoning similar to that which establishes Gauss's first criterion) to coincide with ( — 1 )*'•'' ~^^ e", where e is the unit satisfying the congruence 2T'*-i'=e, mod jp. (ii) If P^p^", |) = 3, modS, there is but one representation of P by (j), and 2( — 1)°+^ = ( — 1)" ; there are 2j' + 1 repre- sentations of P by y] of which two are primitive, 2 (i/ — 1) are derived from the primitive representations of p^, p*, ..., p^^''~'^\ and in the remaining one 6 = 0. Applying Dirichlet's method to the equation in which (r = l, 2, ..., v, /3 = 0, 4a-l-l = ( — l)''g''', and the representation by \f is primitive), we find ( — 1)^ = ( — 1)''; whence inasmuch as the character ( — 1)* is the same in a derived representation, and in the representation from which it is derived 2(-l)» = 1 + 22 (-!)'' = (- 1)'' = 2(-1)«+^. 1 (iii) Lastly, if P=p^'', p = 5, mod 8, there is but one representation by/, and 2 ( — 1)* = + 1 ; there are 2 1/ + 1 representations by (p. Applying Dirichlet's method as in the preceding case, we find that for any primitive representation of an even power of j? by (j>, ( — 1)"+^= +1 ; whence, for a derived repre- sentation in which the greatest common divisor of the indeterminates is g", ( _ ly+e = ( _ i)<r. Consequently 2(-l)«+^ = 2''2 (-l)'' + (-l)''= -M = 2(-l)». This completes the demonstration of Jacobi's theorem. Let P be any uneven number and x (P) the positive numerical transcendent defined by the equation X^ (P) X >f. (P) = 2 ( - 1)JW-') X 2 ( - l)i«i^-') X 2 { - i)|w-i) + *(d^-.), where -^{P) is the number of divisors of P, and d is any divisor of P. It win be seen that x(-P) = 0, except when P is capable of representation both by (j> and f: when P is capable of such simultaneous representation, let P = (4a + l)2-f-16;S* be a representation of P by ^ in which the greatest common divisor of 4a-f-l and /3 is the least possible; let ra- = 4a + l-|-4i,8, and let r ^1 represent the quadratic character (Art. 27) of 1 + i with respect to ;s7 ; the equation 2 ( - 1)» = 2 ( - 1)«+'3 = [11^1 x (P) ss 2 316 REPORT ON THE THEORY OF NUMBERS. [Art. 129. will hold in each of the cases considered separately above ; but the numerical functions occurring in this equation satisfy the condition where Pj, P^ are relatively prime ; the equation is therefore imiversally true for every uneven number P, and implies the identity 2 ( - 1)» o(«»» + l)»+8»» = 2 ( - Ijm + n ^lia + D' + ien' = 2 [—^1 X {P) Q^- From the nature of the identity (2) it is evident that we may substitute any function whatever (which renders the two series convergent) for the exponential of q. Thus, for example, we find [(4m + l)2 + 8n2]i + c [(4to + 1)2 + 16»2]i + p L or J Pi+P = n ^ — - n — ^- — IT — —^ — , (»-iT^,) 1 + where Pi,p2,Ps are primes of the forms 8« + l; 8n + 3; 8n + 5 or 7, respect- ively ; and 17 is a positive or negative unit determined by the congruence (_l)i(i',-i)2i(i'i-i) = ,, mod^i. It would seem that the method of Dirichlet which we have here described may be employed to prove all the theorems of Jacobi's memoir in which the two forms compared have different determinants. Those in which the two forms compared have the same determinant, or determinants differing only by a square factor, are of a more elementary character, and are capable of immediate verification. But Dirichlet's method may also be extended to cases in which one or both of the forms compared has a positive determinant. One example will suffice. If P = (2 a + 1)' + 8 &•' = (2 a + 1)^ - 8 ^\ we have 2 ( - 1)» [{2m + lY + 8«2] = 2 ( -!)•»+» ^ [{2m + 1)^ - 8n^], <p representing any function whatever which renders the series convergent, and the limits of m and n in the first sum being 0, 00, and — oo, + oo ; in the second sum 0, 00, and 1, ^(2m-|-l). 129. We proceed to indicate very briefly the origin of the principal formulae in Jacobi's memoir. Three of them are distinguished from the rest as general, being deduced from the equation (7) of Art. 124, without any specialization. If in that formula we write successively + z and — z, for v, and multiply the results together, the left-hand member becomes 2 ( — l)" 5'"'' + "" z" + " ; m Art. 129.] REPORT ON THE THEORY OF NUMBERS. 317 the right-hand member may be written in the form n{l -5*'»-2)2(l -g*") X II (1-5*'") (1 -2^"'-2z2) (1 -o*'»-2 2-2), 1 1 where the second infinite product, by the equation (7), is equal to 2 ( - 1)" g2m^ 22m and the first to 2 ( - 1)" ^^m^^ Hence 2(-l)'»2'»' + »'2'» + » = 2(-l)'» + " ^^(m' + n^j^am^ ^j^y 1 which is one of Jacobi's general formulae. The other two general formulae, and most of the special ones, are obtained in like manner by considering infinite products which are capable of being expressed in more ways than one as the [product of two Theta functions. To arrive at his special formulae, Jacobi transforms the equation (7) by writing g", where a is positive, for q, and + ^ [for V. He thus obtains the equations 00 n (1 _ Q2ma—a-h\ H _ Q2ma—a + b\ H _ „2)na\ = 2 ( — ']\'»q''"'*^ + '^^ II (1 -j-oZnw-o-*") (l ^Q2ma-a + b'\H _ Q2ma\ _ "^QOnfi + mb^ Any infinite product of either of the types occurring in these equations he calls an elliptic product ; and every infinite product which can be formed in more ways than one by the multiplication of two elliptic products, leads directly to one of his special formulae. The five following elliptic products are of great importance in the theory ; they correspond to the suppositions a = l, 6 = 0; a = 2, 6 = 1; a = |, 6 = ^. n (1 - q^^-^Y n (1 - g^n.) = 2 ( _ 1)™ q^\ n (1 + g2™-i)2 n (1 - g2») = 2 g™', 1 n(l-g2m-i) n(l-g*'») = 2(-l)'»g' 2m'' + m n(i + g2'»-i) n(l-g«"') = 2g 1 2»»2 + m (B) n(l-g™) =2(-l)'»g2(3t»2 + .«); / 1 the first two are the equations (19 and (20) of Art. 124 ; the last is a celebrated formula due to Euler. The infiinite products in the niunerators and denominators of the fractions 318 BEPORT ON THE THEORY OF NUMBERS. [Art. 129. equal to u and u (equations 27 and 28, Art. 125) are all elliptic products of one or other of these five types, in some cases with q^, or (f, or —q substituted for q. Hence a comparison of any two of the fractions equal to « or to m' gives immediately one of Jacobi's special formulae. The demonstration of the formula (2) in Art. 128 will serve as an example of this process. Again, Jacobi has shown that the Eulerian product (Art. 124, E.) n(i+g"')=n ^^,,., . 1 1 ^2 which Euler had himself represented by the fraction 1 1-5*" 2(-i)"'^i<3«.2+m)' \y) can be represented by six other fractions of which both the numerators and denominators are elliptic products ; either the numerator or denominator, or both, being of one of the types (B). Thus, for example, n (1+2—) (1 +?— ) (1 -?-) 2oJ(3«.^.., ?(!+?'") = ^—^ =^^iF^- • (i^) n(l-26»'-3)2(l_28'») ^ ' ^ 1 Here again a comparison of any two of the seven equal fractions gives one of the special formulae : thus writing q^* for q in the two fractions (C) and (D), we find SC — 1)" Q(6m-l)2 + (6n + l)2_2('_l")'n + n g2(6m + l)2 + 2(6»)2 which, however, is only a particular case of the general formula (A). The Eulerian product is also of importance in the theory of the partition of numbers. If it be developed in a series proceeding by powers of q, the coefficient C{m) of the 7nth power of q in the development, expresses the number of ways in which m can be composed by the addition of unequal numbers, or by the addition of equal or unequal uneven numbers. Euler observed that his fractional expression of the product furnishes a recurring formula for the calculation of C(m), and the same thing is true of each of Jacobi's fractions ; the simplest of the seven recurring formulae being that arising from the fraction (D), viz., 2 ( - 1)' C{m -3s') = e, the summation extending to all positive or negative values of s for which m — 3s^ is not negative, and e representing 1, or 0, according as m is or is not of the form ^{Sn^±n). ^^^ Art. 129.] REPORT ON THE THEORY OF NUMBERS. 819 ^^H The equation 5(1 -5'") = ^'(-l)"' g^'sm^ + m) jg memorable historically as ^^^Bthe earliest example of the introduction of a Theta function into analysis *. ^^^^Ut expresses the theorem ^^^^ • The excess of the number of ways in which a given number can be com- V posed by the addition of an even number of unequal numbers above the number ^^^fc of ways in which it can be composed by the addition of an uneven number of ^^^ unequal numbers is ( — 1)" or 0, according as the given number is, or is not, j^ of the form ^(3m- + m).' I^H Of this theorem Jacobi has given an arithmetical demonstration, repro- 1^1 ducing Euler's proof of the analytical formula. ^ 00 1 °° The logarithmic differential of 11 (1 — q^) is 2 $ (m) q"*, where $ (m), as ^in Art. 125, is the sum of the divisors of m : Euler thus obtained the equation 00 +00 +00 2$(m)g''"x 2 (-l)"'gj(3f»2+„)^i 2 (-l)™ + i (3w2 + m)gj'3m2 + »)^ 1 which supplies a recurring formula for the calculation of $ (m), viz., 2(-l)«$(m-i(3s^ + s)) = ^(m), the summation extending to all positive or negative values of s for which m — ^{3s^ + s) is positive, and E{m) representing ( — l)'+im, or 0, according as m is, or is not, of the form ^(Ss^ + s)!- The cube of the Eulerian product is equal to the series i2(-l)"'(2m + l)5z('»'+'»> (Art. 124, equation 22) ; so that [2 (-l)'»52(3'»'+'»)p = i2(-l)'»(2m + 1)52 ("»'+•»', . . . . (F) * In the year 1750 or 1751. Nov. Comm. PetropoL, vol. iii. p. 155. t On the equations n(i +?"■) = n f3^i . n (i -g-) = s (- 1)™ 2*(-''+'»), and their connexion with the partitions and divisors of numbers, see Euler, Nov. Coram. Petropol. vol. iii. p. 125, vol. v. p. 59 and p. 75 ; Acta Petropol. vol. iv. Part I. p. 47 and p. 56 (or Commenta- tionea Arithmeticae Collectae, Nos. IX., XI., XVI., L.; the first memoir in vol. iv. of the Acta is omitted in the collection); Introductio in Analysin Infinitorum, part 4. cap. 16; "Waring, Philo- sophical Transactions for 1788, p. 388 ; Legendre, Th6orie des Nombres, ed. 3, vol. ii. p. 128 ; Jacobi, Fundamenta Nova, p. 185, Crelle, vol. xxxii. p. 164, vol. xvxvii. pp. 67, 73 (or Mathematische Werke, vol. i. p. 345, vol. ii. pp. 73, 79). 320 REPORT ON THE THEORY OF NUMBERS. [Art. 129. a result which in an earlier memoir (Crelle, vol. xxi. p. 13, or translated in Liouville, First Series, vol. vii. p. 85) Jacobi describes as ' hitherto unparalleled in analysis.' Writing q^* for q, and multiplying by q^, it becomes the summations Sj and 2^ extending respectively to all positive uneven numbers, and to all positive uneven numbers prime to 3. In this form it expresses the theorem (3 \ ) extended to all compositions of any number N by the addition of three uneven squares a\, a\, a\, all of which are prime to 3, is ( _ l)i("'-i)m or according as iV is or is not the triple of an uneven square.' Differentiating logaritlimically, we find vet' This equation (in which all the exponents have the same quadratic form) admits of immediate verification, elementary considerations sufficing to show o that the sum 2(— 1)^(*-'> (- j 6(a2_&2^ extended to every solution of the equation i\r=a^ + 3?)^, is zero. Jacobi thus obtains a direct arithmetical proof of the formula (F). (Crelle, vol. xxi. pp. 15-18.) The square and the cube of the Eulerian product can also each of them be represented in two different ways as the quotient of two elliptic products. Other formulae of Jacobi's are inferred from the fundamental equation (7) in a somewhat more complicated way. Replacing v in that equation by certain roots of unity, and multiplying two or more of the results together, Jacobi obtains products which can be expressed in more than one way by means of elliptic products ; the formulae thus deduced are remarkable chiefly because they lead to equations, not between two, but between three or more series, the exponents of which have certain quadratic forms. Lastly, a few additional equalities are derived not from the fundamental equation, but from the modular equations of the third and seventh orders. The modular equation of the third order was brought by Legendre into the form i^kX' + ,s/k\ = 1 ; whence evidently <^2 (ft)) </)2 (3ft,) + vj.2 („) >(,2 (3ft,) = 1 ; writing for the functions (p'^ and •'\f^ their values given by equation (14), Art. 124, Art. 130.] REPORT ON THE THEORY OF NUMBERS. 321 and changing q Into q*, we find 2 (1 — ( — iV' + n) o*("»'' + 3n'^) = 4 So'^^ + l'^ + St^n + l)". I The equation of the seventh order, in the form in which it has been put by M.Gutzlafr*, ^¥x'+^'lcX = h admits of similar treatment, and furnishes as many as seven formulae on account of the variety of expressions which the equations (27) and (28) allow us to substitute for (p and -v/^ in the equation «^(«)<^(7a,) + x|.(«,)x|.(7a,) = l. It is only necessary to observe that we must choose for ^ (w) and ^ (w), and similarly for <p (Jw) and ^ (7 m), expressions having the same denominator. At the beginning of his memoir Jacobi says that the formulae to which it relates are probably finite in number. It would seem that when he expressed himself thus, he had not yet found his three general formulae, each of which contains an infinite number of equations between series having their exponents j contained in the same quadratic form. But it is certainly very unlikely that equations between series whose exponents are contained in different quadratic forms, exist for any but a few of the simplest forms, or for them in infinite number. 130. The Formulae of M. Kronecker. — We now come to an important series of results, discovered within the last few years by M. Kronecker, which form I a memorable accession to our knowledge of quadratic forms, and which have I^^^K opened an entirely new field of arithmetical inquiry. Their demonstration 1^^^ requires considerations of a very complicated kind ; and as they are certainly 1^1 among the most interesting, so also they must be reckoned among the most P^V abstruse of arithmetical truths. Unfortunately, in the brief notices f which * Crelle's Journal, vol. xii. p. 173. t The following are the memoirs of M. Kronecker on the application of the theory of elliptic functions to quadratic forms. (1) 'Ueber elliptische Functionen und Zahlen-Theorie,' Monatsberichte, Oct. 29, 1857; and translated in Liouville, New Series, vol. iii. p. 265. (2) 'Ueber die Anzahl der verschiedenen Klassen von quadratischen Formen von negativer Determinante,' Crelle, vol. Ivii. p. 248 ; and translated in Liouville, vol. v. p. 289. (3) 'Ueber eine nene Eigenachaft der quadratischen Foi-men von negativer Determinante,' Monatsberichte, May 26, 1862. (4) 'Ueber die complexe Multiplication der elUptischer Functionen,' Ibid, June 26, 1862. (5) ' Aufliisung der Pellachen Gleichung mittelst elliptischer Functionem,' Ibid, Jan. 22, 1863. Tt 322 REPORT ON THE THEORY OF NUMBERS. [Art. 130. M. Kronecker has given of his investigations, his methods are indicated only in a very general manner ; and, notwithstanding the light which has been thrown on them in the subsequent memoirs of MM. Hermite and Joubert*, it is occasionally difficult to rediscover them. Nevertheless, as a mere enu- meration of formulae, unaccompanied by any explanation of the methods by which they have been obtained, would be of little use to the reader, we shall attempt in the next article a complete demonstration of one or two of them, which may serve as specimens of the rest. The following (with an unimportant change in the notation) are the eight equations given by M. Kronecker (Crelle, vol. Ivii. p. 248 ; Liouville, New Series, vol. v. p. 289). I. F{2''m) + 2F{2i'm-V) + 2F(2''m-2-') + ... = 2$ (m) + $ (2''-2m) + ^ (2''-2w). II. F{2m) + 2F{2m-V) + 2F{2m-2^) + 2F{2m-3^) + ... = 2$(m). III. F{2m)-2F{2m-V) + 2F(2m-2^)-2F{2m-3^) + ... = 0. IV. 3G(m) + 6G{m-V) + 6G(m~2-') + 6G{m-S^)+... = $ (m) + 3 •*- (m). V. 2F{m) + 4:F{m-l^) + 4.F{m-2^) + 4:F(m-3^)+... = $(m) + Sk(m). VI. 2F{m)-4:F(m-V) + iF{m~2^)-4:F{7n-3^) + ... = {-l)i^^-^'>[^{m)-^{m)]. VII. 2F{m)-4F{m-'i') + 4:F{7n-8^)-'iF{m-12^ + ... = ( _ i)i(«.- 7) $' (^) _ ^' (^)]. VIII. 42.(-l)^('»-')[2i^(^)-3G'(^)] = { - 1)*(«- 1) [$' (m) - ^' (m)]. In these formulae m is any positive uneven number ; in the 1st, m is ^ 2 ; * M. Hermite, ' Sur la thtorie des Equations Modulaires ' ; M. Joubert, ' Sur la Th^orie des ^onctions Elliptiques et son application h, la Theorie des Nombres,' already cited in the note on Art 125. Art. 130.] KEPORT ON THE THEORY OF NUMBERS. 323 in the 7th, m is = - 1, mod 8 ; in the 8th, m is = + 1, mod 8, and the summa- tion extends to all values of s for which iV('^-«^) is integral and not negative ; similarly, the series in the first seven formulae are to be continued until the numbers affected with the signs F and G become negative. If n is any positive number, even or uneven, $ («) is the sum of the divisors of n, ^ (n) the excess of those divisors of n which surpass Vn above those divisors which are surpassed by ^w ; $'(tc) is the sum "^ {-j) d extended to all the divisors of « ; •^' (n) the excess of the sum 2 (-.^ d extended to all the divisors of n which surpass y/n, above the sum 2 (-j^ d extended to all divisors of n which are surpassed by ^i. Lastly, F{n) is the number of uneven classes, G{n) the whole number of classes, of forms of determinant — n ; the classes (1, 0, 1), (1, 1, 1), and their derived classes, being counted as ^ and | respectively ; to F{Qi) we attribute the value 0, to G{Qi) the value —j^*. The arithmetical functions F {n) and G (n) satisfy the equations F{^n) = 2F{n); G{^n) = F{in) + G(n) ; G{n) =F{n), ii' n = l, or 2, mod i ; G{n) = 2F{n),]£n = 7, mod 8 ; G{n) = ^F{n), i£n = 3, mod 8. With the help of these relations (which may be demonstrated by elementary considerations [see Art. 113 of this Report], but which may also be inferred from the theory of elliptic functions) the formulae I. — VIII. may be transformed and combined in various ways, so as to afford new and interesting results. Of these derived formulae M. Kronecker has given two, IX. F{n)+ F{n-1.2) + F{n-2.3) + F{n-3.i) + ... = l^(47i + l). X. E{n) + 2E{n-V) + 2E{n-2^) + 2F{n-S^)+... = |[2 + (-l)»]X(n), * The right-hand members of the formulae I. — VIII. are rendered simpler by this conventional estimation of a class of det. — 1 as J, and of an improperly primitive class of det. — 3 as |^. We have already seen that this convention is a natural one (Art. 126, note); it is, however, less easy to interpret the assumption G {0) ■= — ^. M. Kronecker has given his formulae in their complete expression when these conventional estimations are disregarded ; in his subsequent notes, however, he seems to prefer the simpler form, which we have adopted in the text. T t 2 324 REPORT ON THE THEORY OF NUMBERS, [Art. 130. where n represents any positive integer, X{n) the sum of its uneven divisors, and E{ri) = 2F{n) — G (?i), so that £ {n) is a function satisfying the equations E(in) = E{n); ^(n) = 0, if n = 7, mod 8; E{n) = ^F{n), i£ n = 3, mod 8. The first of these formulae is obtained by subtracting VI. from V. ; for F{n-k.k+l) = ^F(in + l-[2k+lY). The other, if n is uneven, coincides with [V.] — ^ [IV.] ; and with II. if n is unevenly even. If n is the quadruple of an uneven number, its left-hand member may be written in the form [E{in) + 2E{in-V) + 2E{in-2^) + ...] + [^F{n-V) + lFin-S^) + ^,F{n-5^) + ...]; the sima of the first of these series is f X(^w) = f X(ft) ; the second series, coinciding with f [I-]~f [^O' ^^ ^^^ '^^^ ^^^ l-^l*^) ' ^^^ ^^^ series together are therefore equal to 2X{n). Lastly, the formula, if true for any even number, is also true for its quadruple ; for if » = 0, mod 8, E{n) + 2E{n-l') + 2E{n-2^) + ... = [E{in) + 2E{in-V) + 2E{in-2^) + ...] -\-l2E{n-V) + 2E{n-3^) + 2E{n-5'')+...l and every term of the second series is zero, because the numbers n-V, ?i — 3^, ... are all = 7, mod 8. Of the preceding formulae those which contain the functions '^ and ^' are to be regarded as of a more abstruse character than those which only contain X, ^, and 4>'. The latter, in fact, are deducible from known theorems of arithmetic. Thus, if we multiply the formula X. by 12, the right-hand member becomes 8 [2 -f ( — 1)"] X (n), or the number of representations of n as a sum of four squares (see Art. 127). Consequently 12 E {n) is the number of representa- tions of w as a sum of three squares ; for, assuming that this is so for 1 , 2, 3, ..., n — 1, we may infer from the formula X. that it is so for n. Thus the celebrated theorem of Gauss*, which connects the number of representations of a number n as a, sum of three squares, with the number of classes of quadratic forms of det. — n, is contained in the formula X, ; and conversely, that formula is itself deducible from the theorem of Gauss combined with the other and more * Disq. Arith. Art. 291. Legendre had discovered particular cases of the theorem by induction. Hist, de I'Ac. de Paris, 1785, p. 630 sqq. Th6orie des Nombres, ed. 3, vol. i. troisifeme partie. Art. 131.] REPORT ON THE THEORY OF NUMBERS. 325 elementary theorem, which connects the number of representations of n as a sum of four squares with the sum of the uneven divisors of n. 131. Demonstration of the Formulae of M. Kronecker. — We shall first demonstrate the formula V. For this purpose, we consider the equation f(x, 1 - a;) = 0, obtained by writing x for k^, and \—x for X^ in the modular equation f (k^, A^) = of an uneven order n (Art. 125). We shall determine the order of this equation by two different methods ; first, by ascertaining the dimensions of f(x,l— x), when x is increased without limit ; secondly, by I assigning its roots, and the multiplicity of each of them ; a comparison of these two determinations will give the formula V. (i.) Let a; = <^8 (0) ; then because (Art. 125) /.W.(e).x.).n[ x=-^.(l?±?-*)]t, the sign of multiplication IT extending to every system of values of y, y, and h. Let ^ = 1 + -, 0- representing a real positive quantity ; we obtain (Table A, vi. Art. 125). Again, if ^ is the greatest common divisor of y + 2k and y, and if a, h, S' are determined by the equations a- ^ , "- J » - y ?hile c and d are two numbers, of which x is uneven, satisfying the equation ctd — 6c = 1, we find . ,\ C + d ; 5 / , J y den + ay^ 7(l+^) + 2^ ^ a + b ; t Since <f)'{<a + 2) = (f)^{oi) (equation i. Table A, Art. 125), the systems of values represented by are identical, k denoting any term of a complete system of residues, mod y'. 326 REPORT ON THE THEORY OF NUMBERS. [Art. 131. whence (Table A, iii.) if 2l+l = dy, mod S'. If we give to y, y', and k in succession all the <t>(»i) (y(l + i)+2k\ systems of values of which they are susceptible, cp^ \ ; / wQl acquire in succession <I> (») values, which (except for particular values of a) are all different; <p~* ( j? ) will therefore also acquire the same number of different values ; i.e. S will represent in succession every divisor of n, and 21+1 every residue of its conjugate divisor S'. We thus obtain the equation the sign of multiplication extending to every combination of the values of S, S', and 21 + 1. Now let <r increase without limit, so that a;= — yr^—l increases without limit, and is of the same dimensions as e"" (Art. 125). Observing that thefa<!tor ^ « / -x ^ .^S^i + 21 + ly. 8 » has a finite ratio to e"', if 1 ^ ^ , and to e^' , if 1 ^ -^ , we see that the product has a finite ratio to ,l.«*.«l.., that U to r±M]"<"'*"'">. Hence • /« {x, l-x) + x* ^") ■" * ("> is finite, when x increases without Hmit, or^ {x, 1 — x) is of the order ^ (n) + "ir (w). (ii.) Neither nor 1 is a root of the equation fg{x, 1 — x) = ; for /g (0, 1) = 1, /g (1, 0) = 1, therefore (Art. 125) any one of its roots can be represented by (j)* (w), « denoting an imaginary quantity, in which the coeflficient of { is different from zero and positive. But if a; = ^^ (w), 'yw + 2k\ 7 Mx,l-x) = Il[^^{.).<p^C^)]; Art. 131.] BEPOET ON THE THEORY OF NUMBERS. 327 hence the supposition that (p^ (w) is a root of the equation fs{x, 1 — x) = implies that . « /7« + 2^^ for one system (at least) of values of 7, y', and k; i.e. (Art. 125) that there ' satisfying the equation exists a unit matrix and the congruence c, d yw + 2k c + d(i 7 r a-\ a, h c, d = 0,1 1,0 (A) a + bw' , mod 2 (A') Thus, if (p^ (w) is a root of _^ {x, 1 — x) = 0, u> is the root of a quadratic equation 2ak-cy +2 {bJc-^dy +^ay) w + hyco^ = 0, whose extreme coeflBcients are both uneven, and whose determinant, if o- = —hk + ^ {ay + dy'), m o-^ — 7i, a number necessarily negative, because w is imaginary. Conversely, if w is the root of a quadratic equation, of which the extreme coefficients are both uneven, and of which the determinant is negative and included in the formula tr^ — n, <^* («) is a root of _^ {x, 1 — a;) = 0. Or, more precisely, if w is the root of a properly primitive quadratic equation, of which the determinant — A is negative and the extreme coefficients are both uneven, and if n can be repre- sented by the form (1, 0, A) with a positive and uneven value of the second indeterminate, ^* (<o) will be a root of /^{x, l—x) = 0, and the multiplicity of this root will be equal to the number of such representations of n *. To establish this, we shall show (a) that w annuls as many of the factors as there are representations of n ; {^) that ^g (^j 1 ~ ^) is divisible hj x — cp^ (w) as often as there are factora annulled by 00. (a) Let A + 2B00+ Cu)^ = be the equation satisfied by w, and let n = <r^ + ^r^; A, C, and r being positive and uneven ; the four equations 2ak-cy' =tA, hk-\dy' ■\-\ay = TB,\ hy=TC, -hk + ^dy'-^ay = a- ) * The method by which the multiplicity of the roots of the equation f^{x, 1— a;) = is here determined is chiefly taken from M. Joubert's Memoir, ' Sur la Th6orie des Fonctions EUiptiques &c.', pp. 22-24. (B) 328 REPORT ON THE THEORY OF NUMBERS. [Art. 131. will supply one, and only one, system of values for 7, 7', and h, and one, and only ' satisfying the equation (A) and congruence (A'). For the one, unit-raatnx c, d equations ay = TB + <T, by = TC, show that 7 is the greatest common divisor of tB + ct and tC; this common divisor is a divisor of n, because n thus a, h, y, and 7' = - are determined. Again, the congruences 7 2ak = tA 2bk = r^-J"^°^^' determine the value of k, because 2a and 2b have no common divisor with the modulus, while the determinant 2 n 2{-TaB + TbA + o-a) = -[a''-T''B' + T^AC] = 2-=2y is divisible by it ; when k is determined, the equations cy=2ak-TA, dy =2bk-{TB-a), a, b will supply integral values of c and d ; the matrix unit matrix, because c, d thus obtained is a ad-bc= -{ay xdy-byxcy') = -[{<7 + TB){<7-TB) + T^AC] = l; n it also satisfies the congruence a, b c, d 0,1 1,0 mod 2, because, from the equa- tions (B), taken as congruences for the modulus 2, we find b=c = l, mod 2, a = d, mod 2; but also ac? = 0, mod2, so that a=ci? = 0, mod2; lastly, the equation ; — = y— is satisfied by virtue of the first three of the equa- ^ y a + boa -^ ^ tions (B). Thus to each representation of n there corresponds one, and only one, evanescent factor ; conversely to each evanescent factor there corresponds one, and only one, representation of n. For, if w annuls the factor the equation (A) and congruence (A') are satisfied by a unit matrix a, b c, d in which & > 0, but, even if A = — 1, by only one such matrix : so that the equations (B) determine the values of <r and t without ambiguity. The number . Art. 131.] REPORT ON THE THEORY OF NUMBERS. 329 of factors annulled by w is therefore equal to the number of representations of n. (8) Writing ^^ (0) for x, we have /« {x,i-x) = Yi [^^ ie) - cp^ (^,-)] = ^ n, [,|>B (0) - 03 (,)]^ where A is the coefficient of the highest power of a; in _^ (a;, 1 — x), and ITi extends to every root (p^ (w) of fa{x, l—x) = 0, each root having its proper multiplicity. M. Joubert has proved that, if -^^ (6) — 0* (- — -, — ^ vanish when by the usual rule, is neither infinite nor zero. For this limit is, -1- _de r(d) 7lW 6 = , where M is the multiplier appertaining to the transformation w — since (equation 31, Art. 125), yQ-2k n d #.-*-w d ye + 2k> Ldd 6 = > The determination of M is effected as follows : from the equation y(0 + 2k c + d(o or a + bo)' y'c -2ka + (yd -2kh)w — 1 z. ' ya + yboo it appears (Art. 126) that the multiplier corresponding to the compounded transformations y, -2k, y' and a, h c, d , appUed to w, is [a + iTV^y^ ; while that corresponding to the second of these transformations is simply (-1)^^'' ^H (Table A, ii.); therefore ^^ = -((t + itVA)^, and the limit above written u u 330 REPORT ON THE THEORY OF NUMBERS. [Art. 131. becomes — 1 + '^ — ^ /— > which is certainly neither infinite nor zero. Hence <T — It ,^/A f^(x,l—x) is divisible by x — cp^ (w) precisely as often as there are factors '^^{Q) — <p^(- — -, — ) which vanish when 6 = w; i.e. the multiplicity of the root x = (f>^ (to) in /» {x, 1 — x) = is precisely equal t